אינטגרל – ויקיפדיה

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href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><div id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 2024\"\u003Eכאן\u003C/a\u003E.\u003C/b\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="אתר"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="תוכן עניינים" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">התחלה</div> </a> </li> <li id="toc-סימון" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#סימון"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>סימון</span> </div> </a> <ul id="toc-סימון-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-האינטגרל_המסוים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#האינטגרל_המסוים"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>האינטגרל המסוים</span> </div> </a> <button aria-controls="toc-האינטגרל_המסוים-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק האינטגרל המסוים</span> </button> <ul id="toc-האינטגרל_המסוים-sublist" class="vector-toc-list"> <li id="toc-חלוקה_של_קטע" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#חלוקה_של_קטע"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>חלוקה של קטע</span> </div> </a> <ul id="toc-חלוקה_של_קטע-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הגדרת_האינטגרל_המסוים_באמצעות_סכומי_רימן" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הגדרת_האינטגרל_המסוים_באמצעות_סכומי_רימן"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>הגדרת האינטגרל המסוים באמצעות סכומי רימן</span> </div> </a> <ul id="toc-הגדרת_האינטגרל_המסוים_באמצעות_סכומי_רימן-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הגדרה_באמצעות_סכומי_דארבו" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הגדרה_באמצעות_סכומי_דארבו"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>הגדרה באמצעות סכומי דארבו</span> </div> </a> <ul id="toc-הגדרה_באמצעות_סכומי_דארבו-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-מרחב_הפונקציות_האינטגרביליות" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#מרחב_הפונקציות_האינטגרביליות"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>מרחב הפונקציות האינטגרביליות</span> </div> </a> <ul id="toc-מרחב_הפונקציות_האינטגרביליות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-חישוב_האינטגרל_המסוים" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#חישוב_האינטגרל_המסוים"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>חישוב האינטגרל המסוים</span> </div> </a> <ul id="toc-חישוב_האינטגרל_המסוים-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-האינטגרל_הלא_מסוים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#האינטגרל_הלא_מסוים"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>האינטגרל הלא מסוים</span> </div> </a> <button aria-controls="toc-האינטגרל_הלא_מסוים-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק האינטגרל הלא מסוים</span> </button> <ul id="toc-האינטגרל_הלא_מסוים-sublist" class="vector-toc-list"> <li id="toc-הגדרה" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#הגדרה"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>הגדרה</span> </div> </a> <ul id="toc-הגדרה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-דוגמה" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#דוגמה"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>דוגמה</span> </div> </a> <ul id="toc-דוגמה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-מציאת_האינטגרל_הלא_מסוים" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#מציאת_האינטגרל_הלא_מסוים"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>מציאת האינטגרל הלא מסוים</span> </div> </a> <ul id="toc-מציאת_האינטגרל_הלא_מסוים-sublist" class="vector-toc-list"> <li id="toc-הערות" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#הערות"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>הערות</span> </div> </a> <ul id="toc-הערות-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-הכללות_של_אינטגרל_רימן" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הכללות_של_אינטגרל_רימן"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>הכללות של אינטגרל רימן</span> </div> </a> <button aria-controls="toc-הכללות_של_אינטגרל_רימן-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>שינוי מצב התת־פרק הכללות של אינטגרל רימן</span> </button> <ul id="toc-הכללות_של_אינטגרל_רימן-sublist" class="vector-toc-list"> <li id="toc-אינטגרל_לבג" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#אינטגרל_לבג"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>אינטגרל לבג</span> </div> </a> <ul id="toc-אינטגרל_לבג-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-אינטגרל_רימן־סטילטיס_ואינטגרל_לבג־סטילטיס" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#אינטגרל_רימן־סטילטיס_ואינטגרל_לבג־סטילטיס"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>אינטגרל רימן־סטילטיס ואינטגרל לבג־סטילטיס</span> </div> </a> <ul id="toc-אינטגרל_רימן־סטילטיס_ואינטגרל_לבג־סטילטיס-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-שימושי_האינטגרל" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#שימושי_האינטגרל"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>שימושי האינטגרל</span> </div> </a> <ul id="toc-שימושי_האינטגרל-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-נוסחאות_אינטגרציה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#נוסחאות_אינטגרציה"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>נוסחאות אינטגרציה</span> </div> </a> <ul id="toc-נוסחאות_אינטגרציה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ראו_גם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ראו_גם"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-הערות_שוליים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הערות_שוליים"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>הערות שוליים</span> </div> </a> <ul id="toc-הערות_שוליים-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="תוכן עניינים" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="מצב תוכן העניינים" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">מצב תוכן העניינים</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">אינטגרל</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="מעבר לערך בשפה אחרת. זמין ב־92 שפות" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-92" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">92 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integral" title="Integral – אנגלית" lang="en" hreflang="en" data-title="Integral" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%8C%A0%E1%88%AB%E1%89%83%E1%88%9A" title="አጠራቃሚ – אמהרית" lang="am" hreflang="am" data-title="አጠራቃሚ" data-language-autonym="አማርኛ" data-language-local-name="אמהרית" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – אראגונית" lang="an" hreflang="an" data-title="Integración" data-language-autonym="Aragonés" data-language-local-name="אראגונית" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%83%D8%A7%D9%85%D9%84" title="تكامل – ערבית" lang="ar" hreflang="ar" data-title="تكامل" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – אסטורית" lang="ast" hreflang="ast" data-title="Integración" data-language-autonym="Asturianu" data-language-local-name="אסטורית" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0nteqral" title="İnteqral – אזרית" lang="az" hreflang="az" data-title="İnteqral" data-language-autonym="Azərbaycanca" data-language-local-name="אזרית" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D9%82%D8%B1%D8%A7%D9%84" title="انتقرال – אזרבייג׳נית דרומית" lang="azb" hreflang="azb" data-title="انتقرال" data-language-autonym="تۆرکجه" data-language-local-name="אזרבייג׳נית דרומית" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – בשקירית" lang="ba" hreflang="ba" data-title="Интеграл" data-language-autonym="Башҡортса" data-language-local-name="בשקירית" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D1%8D%D0%B3%D1%80%D0%B0%D0%BB" title="Інтэграл – בלארוסית" lang="be" hreflang="be" data-title="Інтэграл" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – בולגרית" lang="bg" hreflang="bg" data-title="Интеграл" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AF%E0%A7%8B%E0%A6%97%E0%A6%9C%E0%A7%80%E0%A6%95%E0%A6%B0%E0%A6%A3" title="যোগজীকরণ – בנגלית" lang="bn" hreflang="bn" data-title="যোগজীকরণ" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Integral" title="Integral – בוסנית" lang="bs" hreflang="bs" data-title="Integral" data-language-autonym="Bosanski" data-language-local-name="בוסנית" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="ערך מומלץ"><a href="https://ca.wikipedia.org/wiki/Integraci%C3%B3" title="Integració – קטלאנית" lang="ca" hreflang="ca" data-title="Integració" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%95%D9%88%D8%A7%D9%88%DA%A9%D8%A7%D8%B1%DB%8C" title="تەواوکاری – כורדית סוראנית" lang="ckb" hreflang="ckb" data-title="تەواوکاری" data-language-autonym="کوردی" data-language-local-name="כורדית סוראנית" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Integr%C3%A1l" title="Integrál – צ׳כית" lang="cs" hreflang="cs" data-title="Integrál" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – צ׳ובשית" lang="cv" hreflang="cv" data-title="Интеграл" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Integryn" title="Integryn – ולשית" lang="cy" hreflang="cy" data-title="Integryn" data-language-autonym="Cymraeg" data-language-local-name="ולשית" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Integral" title="Integral – גרמנית" lang="de" hreflang="de" data-title="Integral" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/%C4%B0ntegral" title="İntegral – זזקית" lang="diq" hreflang="diq" data-title="İntegral" data-language-autonym="Zazaki" data-language-local-name="זזקית" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-dtp mw-list-item"><a href="https://dtp.wikipedia.org/wiki/Ponompuuvan" title="Ponompuuvan – דוסונית" lang="dtp" hreflang="dtp" data-title="Ponompuuvan" data-language-autonym="Kadazandusun" data-language-local-name="דוסונית" class="interlanguage-link-target"><span>Kadazandusun</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BB%CE%BF%CE%BA%CE%BB%CE%AE%CF%81%CF%89%CE%BC%CE%B1" title="Ολοκλήρωμα – יוונית" lang="el" hreflang="el" data-title="Ολοκλήρωμα" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Integralo" title="Integralo – אספרנטו" lang="eo" hreflang="eo" data-title="Integralo" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Integraci%C3%B3n" title="Integración – ספרדית" lang="es" hreflang="es" data-title="Integración" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Integraal" title="Integraal – אסטונית" lang="et" hreflang="et" data-title="Integraal" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu badge-Q17437796 badge-featuredarticle mw-list-item" title="ערך מומלץ"><a href="https://eu.wikipedia.org/wiki/Integral" title="Integral – בסקית" lang="eu" hreflang="eu" data-title="Integral" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%DA%AF%D8%B1%D8%A7%D9%84" title="انتگرال – פרסית" lang="fa" hreflang="fa" data-title="انتگرال" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Integraali" title="Integraali – פינית" lang="fi" hreflang="fi" data-title="Integraali" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Int%C3%A9gration_(math%C3%A9matiques)" title="Intégration (mathématiques) – צרפתית" lang="fr" hreflang="fr" data-title="Intégration (mathématiques)" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Integral" title="Integral – גליסית" lang="gl" hreflang="gl" data-title="Integral" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%B8%E0%AA%82%E0%AA%95%E0%AA%B2%E0%AA%A8" title="સંકલન – גוג׳ארטי" lang="gu" hreflang="gu" data-title="સંકલન" data-language-autonym="ગુજરાતી" data-language-local-name="גוג׳ארטי" class="interlanguage-link-target"><span>ગુજરાતી</span></a></li><li class="interlanguage-link interwiki-hak mw-list-item"><a href="https://hak.wikipedia.org/wiki/Chit-f%C3%BBn-ho%CC%8Dk" title="Chit-fûn-ho̍k – סינית האקה" lang="hak" hreflang="hak" data-title="Chit-fûn-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="סינית האקה" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A4%BE%E0%A4%95%E0%A4%B2%E0%A4%A8" title="समाकलन – הינדי" lang="hi" hreflang="hi" data-title="समाकलन" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Integral" title="Integral – קרואטית" lang="hr" hreflang="hr" data-title="Integral" data-language-autonym="Hrvatski" data-language-local-name="קרואטית" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Entegrasyon_(matematik)" title="Entegrasyon (matematik) – קראולית האיטית" lang="ht" hreflang="ht" data-title="Entegrasyon (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="קראולית האיטית" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Integr%C3%A1l" title="Integrál – הונגרית" lang="hu" hreflang="hu" data-title="Integrál" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B6%D5%BF%D5%A5%D5%A3%D6%80%D5%A1%D5%AC" title="Ինտեգրալ – ארמנית" lang="hy" hreflang="hy" data-title="Ինտեգրալ" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Integral" title="Integral – אינדונזית" lang="id" hreflang="id" data-title="Integral" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Integralo" title="Integralo – אידו" lang="io" hreflang="io" data-title="Integralo" data-language-autonym="Ido" data-language-local-name="אידו" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Heildun" title="Heildun – איסלנדית" lang="is" hreflang="is" data-title="Heildun" data-language-autonym="Íslenska" data-language-local-name="איסלנדית" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Integrale" title="Integrale – איטלקית" lang="it" hreflang="it" data-title="Integrale" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9C%E1%83%A2%E1%83%94%E1%83%92%E1%83%A0%E1%83%90%E1%83%9A%E1%83%98" title="ინტეგრალი – גאורגית" lang="ka" hreflang="ka" data-title="ინტეგრალი" data-language-autonym="ქართული" data-language-local-name="גאורגית" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Integral" title="Integral – קרקלפקית" lang="kaa" hreflang="kaa" data-title="Integral" data-language-autonym="Qaraqalpaqsha" data-language-local-name="קרקלפקית" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – קזחית" lang="kk" hreflang="kk" data-title="Интеграл" data-language-autonym="Қазақша" data-language-local-name="קזחית" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%A2%E1%9E%B6%E1%9F%86%E1%9E%84%E1%9E%8F%E1%9F%81%E1%9E%80%E1%9F%92%E1%9E%9A%E1%9E%B6%E1%9E%9B" title="អាំងតេក្រាល – קמרית" lang="km" hreflang="km" data-title="អាំងតេក្រាល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="קמרית" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%85%E0%B2%A8%E0%B3%81%E0%B2%95%E0%B2%B2%E0%B2%A8%E0%B2%B6%E0%B2%BE%E0%B2%B8%E0%B3%8D%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ಅನುಕಲನಶಾಸ್ತ್ರ – קנאדה" lang="kn" hreflang="kn" data-title="ಅನುಕಲನಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="קנאדה" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%81%EB%B6%84" title="적분 – קוריאנית" lang="ko" hreflang="ko" data-title="적분" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/%C3%8Entegral" title="Întegral – כורדית כורמנג׳ית" lang="ku" hreflang="ku" data-title="Întegral" data-language-autonym="Kurdî" data-language-local-name="כורדית כורמנג׳ית" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – קירגיזית" lang="ky" hreflang="ky" data-title="Интеграл" data-language-autonym="Кыргызча" data-language-local-name="קירגיזית" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Integrale" title="Integrale – לטינית" lang="la" hreflang="la" data-title="Integrale" data-language-autonym="Latina" data-language-local-name="לטינית" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Integral" title="Integral – לומברדית" lang="lmo" hreflang="lmo" data-title="Integral" data-language-autonym="Lombard" data-language-local-name="לומברדית" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Integralas" title="Integralas – ליטאית" lang="lt" hreflang="lt" data-title="Integralas" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Integr%C4%81lis" title="Integrālis – לטבית" lang="lv" hreflang="lv" data-title="Integrālis" data-language-autonym="Latviešu" data-language-local-name="לטבית" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%BE%E0%B4%95%E0%B4%B2%E0%B4%A8%E0%B4%82" title="സമാകലനം – מליאלאם" lang="ml" hreflang="ml" data-title="സമാകലനം" data-language-autonym="മലയാളം" data-language-local-name="מליאלאם" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – מונגולית" lang="mn" hreflang="mn" data-title="Интеграл" data-language-autonym="Монгол" data-language-local-name="מונגולית" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%95%E0%A4%B2%E0%A4%A8" title="संकलन – מראטהית" lang="mr" hreflang="mr" data-title="संकलन" data-language-autonym="मराठी" data-language-local-name="מראטהית" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kamiran" title="Kamiran – מלאית" lang="ms" hreflang="ms" data-title="Kamiran" data-language-autonym="Bahasa Melayu" data-language-local-name="מלאית" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/L-Integral" title="L-Integral – מלטית" lang="mt" hreflang="mt" data-title="L-Integral" data-language-autonym="Malti" data-language-local-name="מלטית" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%84%E1%80%BA%E1%80%90%E1%80%AE%E1%80%82%E1%80%9B%E1%80%B1%E1%80%B8%E1%80%9B%E1%80%BE%E1%80%84%E1%80%BA%E1%80%B8" title="အင်တီဂရေးရှင်း – בורמזית" lang="my" hreflang="my" data-title="အင်တီဂရေးရှင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="בורמזית" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl badge-Q70894304 mw-list-item" title=""><a href="https://nl.wikipedia.org/wiki/Integraal" title="Integraal – הולנדית" lang="nl" hreflang="nl" data-title="Integraal" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Integral" title="Integral – נורווגית חדשה" lang="nn" hreflang="nn" data-title="Integral" data-language-autonym="Norsk nynorsk" data-language-local-name="נורווגית חדשה" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Integrasjon" title="Integrasjon – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Integrasjon" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Integracion" title="Integracion – אוקסיטנית" lang="oc" hreflang="oc" data-title="Integracion" data-language-autonym="Occitan" data-language-local-name="אוקסיטנית" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Wajjummaa" title="Wajjummaa – אורומו" lang="om" hreflang="om" data-title="Wajjummaa" data-language-autonym="Oromoo" data-language-local-name="אורומו" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Ca%C5%82ka" title="Całka – פולנית" lang="pl" hreflang="pl" data-title="Całka" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%86%D9%B9%DB%8C%DA%AF%D8%B1%D9%84" title="انٹیگرل – פנג׳בית מערבית" lang="pnb" hreflang="pnb" data-title="انٹیگرل" data-language-autonym="پنجابی" data-language-local-name="פנג׳בית מערבית" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Integral" title="Integral – פורטוגזית" lang="pt" hreflang="pt" data-title="Integral" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Integral%C4%83" title="Integrală – רומנית" lang="ro" hreflang="ro" data-title="Integrală" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – רוסית" lang="ru" hreflang="ru" data-title="Интеграл" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Intiggrali" title="Intiggrali – סיציליאנית" lang="scn" hreflang="scn" data-title="Intiggrali" data-language-autonym="Sicilianu" data-language-local-name="סיציליאנית" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Integral" title="Integral – סקוטית" lang="sco" hreflang="sco" data-title="Integral" data-language-autonym="Scots" data-language-local-name="סקוטית" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Integral" title="Integral – סרבו-קרואטית" lang="sh" hreflang="sh" data-title="Integral" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="סרבו-קרואטית" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Integral" title="Integral – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Integral" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Integr%C3%A1l" title="Integrál – סלובקית" lang="sk" hreflang="sk" data-title="Integrál" data-language-autonym="Slovenčina" data-language-local-name="סלובקית" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Integral" title="Integral – סלובנית" lang="sl" hreflang="sl" data-title="Integral" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Integrali" title="Integrali – אלבנית" lang="sq" hreflang="sq" data-title="Integrali" data-language-autonym="Shqip" data-language-local-name="אלבנית" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – סרבית" lang="sr" hreflang="sr" data-title="Интеграл" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Integral" title="Integral – סונדנזית" lang="su" hreflang="su" data-title="Integral" data-language-autonym="Sunda" data-language-local-name="סונדנזית" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Integral" title="Integral – שוודית" lang="sv" hreflang="sv" data-title="Integral" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Ukamilishaji_(hisabati)" title="Ukamilishaji (hisabati) – סווהילי" lang="sw" hreflang="sw" data-title="Ukamilishaji (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="סווהילי" class="interlanguage-link-target"><span>Kiswahili</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%8A%E0%AE%95%E0%AF%88%E0%AE%AF%E0%AF%80%E0%AE%9F%E0%AF%81" title="தொகையீடு – טמילית" lang="ta" hreflang="ta" data-title="தொகையீடு" data-language-autonym="தமிழ்" data-language-local-name="טמילית" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9B%E0%B8%A3%E0%B8%B4%E0%B8%9E%E0%B8%B1%E0%B8%99%E0%B8%98%E0%B9%8C" title="ปริพันธ์ – תאית" lang="th" hreflang="th" data-title="ปริพันธ์" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0ntegral" title="İntegral – טורקית" lang="tr" hreflang="tr" data-title="İntegral" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Интеграл – טטרית" lang="tt" hreflang="tt" data-title="Интеграл" data-language-autonym="Татарча / tatarça" data-language-local-name="טטרית" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB" title="Інтеграл – אוקראינית" lang="uk" hreflang="uk" data-title="Інтеграл" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%DA%A9%D8%A7%D9%85%D9%84" title="تکامل – אורדו" lang="ur" hreflang="ur" data-title="تکامل" data-language-autonym="اردو" data-language-local-name="אורדו" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Integral" title="Integral – אוזבקית" lang="uz" hreflang="uz" data-title="Integral" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="אוזבקית" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Integral" title="Integral – ונציאנית" lang="vec" hreflang="vec" data-title="Integral" data-language-autonym="Vèneto" data-language-local-name="ונציאנית" class="interlanguage-link-target"><span>Vèneto</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADch_ph%C3%A2n" title="Tích phân – וייטנאמית" lang="vi" hreflang="vi" data-title="Tích phân" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wa mw-list-item"><a href="https://wa.wikipedia.org/wiki/Riveye" title="Riveye – ולונית" lang="wa" hreflang="wa" data-title="Riveye" data-language-autonym="Walon" data-language-local-name="ולונית" class="interlanguage-link-target"><span>Walon</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A7%AF%E5%88%86" title="积分 – סינית" lang="zh" hreflang="zh" data-title="积分" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Chek-hun" title="Chek-hun – מין נאנית" lang="nan" hreflang="nan" data-title="Chek-hun" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="מין נאנית" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%A9%8D%E5%88%86" title="積分 – קנטונזית" lang="yue" hreflang="yue" data-title="積分" data-language-autonym="粵語" data-language-local-name="קנטונזית" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q80091#sitelinks-wikipedia" title="עריכת קישורים בין־לשוניים" class="wbc-editpage">עריכת הקישורים</a></span></div> </div> 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Disambig_RTL.svg/38px-Disambig_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Disambig_RTL.svg/50px-Disambig_RTL.svg.png 2x" data-file-width="220" data-file-height="168" /></span></span> ערך זה עוסק בפעולה מתמטית. אם התכוונתם למשמעות אחרת, ראו <span class="nodisambig"><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" class="mw-disambig" title="אינטגרציה">אינטגרציה</a></span>.</div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Integral_as_region_under_curve.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Integral_as_region_under_curve.png/250px-Integral_as_region_under_curve.png" decoding="async" width="250" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Integral_as_region_under_curve.png/375px-Integral_as_region_under_curve.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/90/Integral_as_region_under_curve.png/500px-Integral_as_region_under_curve.png 2x" data-file-width="782" data-file-height="685" /></a><figcaption>עבור פונקציה חיובית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle f(x)}"></span>, האינטגרל המסוים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.139ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx}"></span> הוא השטח <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10a3c52d186162ec8910ebc0288ce982aef842f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\textstyle S}"></span> הכלוא מתחת לגרף הפונקציה.</figcaption></figure> <p><b>אִינְטֶגְרָל</b> או <b>אַסְכֶּמֶת</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> הוא מושג <a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה">מתמטי</a> בתחום ה<a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון האינפיניטסימלי</a>, המהווה (עבור <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%9E%D7%A9%D7%99%D7%AA" title="פונקציה ממשית">פונקציה ממשית</a>) <a href="/wiki/%D7%94%D7%9B%D7%9C%D7%9C%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="הכללה (מתמטיקה)">הכללה מתמטית</a> של מושג ה<a href="/wiki/%D7%A1%D7%9B%D7%95%D7%9D" title="סכום">סכום</a>. לאינטגרל שימושים רבים ביותר, וּבהם חישוב <a href="/wiki/%D7%A9%D7%98%D7%97" title="שטח">שטח</a> של תחום מישורי, <a href="/wiki/%D7%A0%D7%A4%D7%97" title="נפח">נפח</a> של מרחב <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%91-%D7%9E%D7%9E%D7%93%D7%99" title="אינטגרל רב-ממדי">רב־ממדי</a>, <a href="/wiki/%D7%9E%D7%A1%D7%94" title="מסה">מסה</a> של גוף, <a href="/wiki/%D7%90%D7%95%D7%A8%D7%9A" title="אורך">אורך</a> של מסילה עקומה, <a href="/wiki/%D7%94%D7%A1%D7%AA%D7%91%D7%A8%D7%95%D7%AA" title="הסתברות">הסתברות</a> של משתנים מקריים רציפים, <a href="/wiki/%D7%9B%D7%95%D7%97_(%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94)" title="כוח (פיזיקה)">כוח</a> הפועל בין שני גופים, <a href="/wiki/%D7%90%D7%A0%D7%A8%D7%92%D7%99%D7%99%D7%AA_%D7%97%D7%95%D7%9D" class="mw-redirect" title="אנרגיית חום">אנרגיית החום</a> הכוללת של אמבט, <a href="/wiki/%D7%9E%D7%94%D7%99%D7%A8%D7%95%D7%AA" title="מהירות">מהירות</a> מקומו המרחבי של גוף הנע בהשפעת כוח בעל עצמה משתנה ועוד. </p><p>המושג הכללי של אינטגרל כולל בתוכו שני מושגים שונים לכאורה: האינטגרל המסוים והאינטגרל הלא־מסוים (קרי: הפונקציה הקדומה). </p> <ul><li><b>האינטגרל המסוים</b> של פונקציה אי־שלילית המוגדרת על <a href="/wiki/%D7%A7%D7%98%D7%A2" class="mw-redirect" title="קטע">קטע</a> סופי, הוא מספר השווה ל<a href="/wiki/%D7%A9%D7%98%D7%97" title="שטח">שטח</a> הכלוא בין ציר ה־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> לבין <a href="/wiki/%D7%92%D7%A8%D7%A3_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גרף של פונקציה">גרף הפונקציה</a>, בין קצות הקטע (ראו תרשים משמאל).</li> <li><b>האינטגרל הלא־מסוים</b> או <b>הפונקציה הקדומה</b> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span> מציינים את קבוצת כל הפונקציות הממשיות, ש<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרתן</a> שווה ל־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span>. לפונקציה מסוג זה נהוג לקרוא "<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%93%D7%95%D7%9E%D7%94" class="mw-redirect" title="פונקציה קדומה">פונקציה קדומה</a>" של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span>.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:%D0%A7%D1%82%D0%BE_%D1%82%D0%B0%D0%BA%D0%BE%D0%B5_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB_%D0%90%D0%BD%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/%D0%A7%D1%82%D0%BE_%D1%82%D0%B0%D0%BA%D0%BE%D0%B5_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB_%D0%90%D0%BD%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F.gif/220px-%D0%A7%D1%82%D0%BE_%D1%82%D0%B0%D0%BA%D0%BE%D0%B5_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB_%D0%90%D0%BD%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F.gif" decoding="async" width="220" height="258" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/c0/%D0%A7%D1%82%D0%BE_%D1%82%D0%B0%D0%BA%D0%BE%D0%B5_%D0%B8%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB_%D0%90%D0%BD%D0%B8%D0%BC%D0%B0%D1%86%D0%B8%D1%8F.gif 1.5x" data-file-width="300" data-file-height="352" /></a><figcaption>מהו אינטגרל (אנימציה)</figcaption></figure> <p><a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a> קובע ששני המושגים הללו מתלכדים. כלומר: אם הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span> אינטגרבילית בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span> (בהמשך יוגדרו התנאים לאינטגרביליות) וגם יש לה פונקציה קדומה, אז האינטגרל המסוים של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span> בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span> שווה לביטוי <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F(b)-F(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F(b)-F(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3f460117d5265fec9fb458f74396f3dd96a3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.168ex; height:2.843ex;" alt="{\textstyle F(b)-F(a)}"></span>, כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8256dae10b9e3abb3592ff608e81c8bc324edce3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\textstyle F}"></span> מסמנת את הפונקציה הקדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b77076edca76caf3331d0551d1645b8f678283" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}"></span>. </p> <table cellpadding="1" style="float: left; clear:both; border: 1px solid #8888aa; background: #f7f8ff; padding: 5px; font-size: 95%; margin: 0.5em 1em 0.5em 0.5em;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/60px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/90px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <p>בערך זה<br />נעשה שימוש<br />בסימנים מוסכמים<br />מתחום המתמטיקה.<br />להבהרת הסימנים<br />ראו <a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי">סימון מתמטי</a>. </p> </td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="סימון"><span id=".D7.A1.D7.99.D7.9E.D7.95.D7.9F"></span>סימון</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=1" title="עריכת קוד המקור של הפרק: סימון"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=1" title="עריכת פסקה: "סימון"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>את האינטגרל מסמנים בסימן ∫ שניתן על ידי <a href="/wiki/%D7%92%D7%95%D7%98%D7%A4%D7%A8%D7%99%D7%93_%D7%95%D7%99%D7%9C%D7%94%D7%9C%D7%9D_%D7%9C%D7%99%D7%99%D7%91%D7%A0%D7%99%D7%A5" title="גוטפריד וילהלם לייבניץ">גוטפריד וילהלם לייבניץ</a> ושמקורו ב־<a href="/wiki/S_%D7%90%D7%A8%D7%95%D7%9B%D7%94" title="S ארוכה">S הארוכה</a> שבתחילת המילה הלטינית Summa (סְכוּם), שאותה הוא כתב כ־ſumma. אין לבלבל בינו לבין האות ʃ, המייצגת <a href="/wiki/%D7%A2%D7%99%D7%A6%D7%95%D7%A8_%D7%91%D7%AA%D7%A8-%D7%9E%D7%9B%D7%AA%D7%A9%D7%99,_%D7%97%D7%95%D7%9B%D7%9A_%D7%A9%D7%95%D7%A8%D7%A7,_%D7%90%D7%98%D7%95%D7%9D" title="עיצור בתר-מכתשי, חוכך שורק, אטום">עיצור בתר-מכתשי, חוכך שורק, אטום (שׁ)</a> ב<a href="/wiki/%D7%90%D7%9C%D7%A4%D7%91%D7%99%D7%AA_%D7%A4%D7%95%D7%A0%D7%98%D7%99_%D7%91%D7%99%D7%A0%D7%9C%D7%90%D7%95%D7%9E%D7%99" class="mw-redirect" title="אלפבית פונטי בינלאומי">אלפבית הפונטי הבינלאומי</a>. </p> <div class="mw-heading mw-heading2"><h2 id="האינטגרל_המסוים"><span id=".D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.94.D7.9E.D7.A1.D7.95.D7.99.D7.9D"></span>האינטגרל המסוים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=2" title="עריכת קוד המקור של הפרק: האינטגרל המסוים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=2" title="עריכת פסקה: "האינטגרל המסוים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>כאמור לעיל, האינטגרל המסוים של פונקציה ממשית, מעל קטע סגור, שווה לשטח שמתחת לגרף הפונקציה ומעל ציר ה־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> (כאשר השטח שמתחת לציר ה־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> מוגדר כשטח שלילי). כדי לקבל הגדרה מסודרת של האינטגרל, יש לבחון את המשמעות המדויקת של המושג "שטח", במיוחד כאשר הגרף אינו ישר, ואף אינו בהכרח <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="פונקציה רציפה (אנליזה)">רציף</a>. השטח של <a href="/wiki/%D7%9E%D7%9C%D7%91%D7%9F" title="מלבן">מלבן</a> מובן היטב, ואחת הדרכים להגדיר את השטח שמתחת לגרף היא לקרב את הצורה שמתחת לגרף באמצעות מלבנים זרים (וּמקבילים לצירים). בהגדרה זו יש בעיה עקרונית: כל עוד מספר המלבנים סופי, והפונקציה אינה מלבנית, השטח הכולל שלהם אינו אלא קירוב של השטח האמיתי, וניתן לשפר את הקירוב על ידי הקטנת המלבנים והגדלת מספרם. </p><p>ברנרד רימן הפך את הבעיה הזו להגדרה של ערך האינטגרל: במקום לחפש את הקירוב הטוב ביותר, שלרוב אינו קיים, מסכמים מלבנים קטנים ורבים יותר, כך שהקירוב ילך וישתפר. בסופו של דבר בוחרים את ה<a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" title="גבול של סדרה">גבול</a> של סדרת הסכומים המתקבלת מסדרת הקירובים. גישה דומה, המגיעה לאותן תוצאות, מבוססת על <b>סכומי דארבו</b>: כאן מקרבים את הצורה החוסמת מלמטה ומלמעלה באמצעות מלבנים הכלואים מתחת לגרף ומלבנים הכולאים אותו מלמעלה. </p><p>הראשון שהגדיר את האינטגרל המסוים במדויק היה <a href="/wiki/%D7%91%D7%A8%D7%A0%D7%94%D7%A8%D7%93_%D7%A8%D7%99%D7%9E%D7%9F" title="ברנהרד רימן">רימן</a>. בעקבותיו הוצעו הכללות רבות: ב<a href="/wiki/%D7%AA%D7%97%D7%95%D7%9D_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94#תחום_ההגדרה" title="תחום של פונקציה">תחום של פונקציה#תחום ההגדרה</a> של הפונקציה (העשוי להיות <a href="/wiki/%D7%94%D7%99%D7%A9%D7%A8_%D7%94%D7%9E%D7%9E%D7%A9%D7%99" title="הישר הממשי">הישר הממשי</a> כולו, קובייה ב<a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%90%D7%95%D7%A7%D7%9C%D7%99%D7%93%D7%99" title="מרחב אוקלידי">מרחב האוקלידי</a> ה־n־ממדי, או כל <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%A7%D7%95%D7%9E%D7%A4%D7%A7%D7%98%D7%99_%D7%9E%D7%A7%D7%95%D7%9E%D7%99%D7%AA" title="מרחב קומפקטי מקומית">מרחב קומפקטי מקומית</a>), בערכים שהפונקציה עשויה לקבל (מספרים ממשיים או מרוכבים, <a href="/wiki/%D7%9E%D7%A1%D7%A4%D7%A8_p-%D7%90%D7%93%D7%99" title="מספר p-אדי">מספרים p־אדיים</a>, או <a href="/wiki/%D7%95%D7%A7%D7%98%D7%95%D7%A8_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" class="mw-redirect" title="וקטור (אלגברה)">וקטורים</a> שאלו רכיביהם), ובאופי הפונקציות שעבורן מחושב האינטגרל. </p> <div class="mw-heading mw-heading3"><h3 id="חלוקה_של_קטע"><span id=".D7.97.D7.9C.D7.95.D7.A7.D7.94_.D7.A9.D7.9C_.D7.A7.D7.98.D7.A2"></span>חלוקה של קטע</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=3" title="עריכת קוד המקור של הפרק: חלוקה של קטע"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=3" title="עריכת פסקה: "חלוקה של קטע"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>סדרה (סופית) של נקודות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99c2af530a82249e7bb239caf9bc5193ee766bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.506ex; height:2.509ex;" alt="{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}"></span> נקראת <b>חלוקה</b> של הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span>. את הסדרה אפשר לפרש כאילו היא מחלקת את הקטע ל־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span> תת־קטעים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [x_{0},x_{1}],[x_{1},x_{2}],\dotsc ,[x_{n-1},x_{n}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [x_{0},x_{1}],[x_{1},x_{2}],\dotsc ,[x_{n-1},x_{n}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f07c76e2579426a368f574157d06df4a55a96b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.928ex; height:2.843ex;" alt="{\textstyle [x_{0},x_{1}],[x_{1},x_{2}],\dotsc ,[x_{n-1},x_{n}]}"></span>, ה<a href="/wiki/%D7%97%D7%99%D7%AA%D7%95%D7%9A_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חיתוך (מתמטיקה)">חותכים</a> זה את זה רק בנקודות הקצה שלהם. לכל חלוקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d9f6a29fcb6e9c0c5d94804f9ee4b271e621b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\textstyle \pi }"></span> אפשר להגדיר את <b>קוטר</b> החלוקה (או <b>פרמטר החלוקה</b>) באופן הבא: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (\pi )=\max _{1\leq i\leq n}|x_{i}-x_{i-1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda (\pi )=\max _{1\leq i\leq n}|x_{i}-x_{i-1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2babb6406bff8e560676230bcc03d1ce405f93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.408ex; height:4.176ex;" alt="{\displaystyle \lambda (\pi )=\max _{1\leq i\leq n}|x_{i}-x_{i-1}|}"></span>. מכיוון שסכום אורכי הקטעים הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle b-a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd65e7abcecaa69defef3289f788789ee52aa6da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\textstyle b-a}"></span>, הקוטר של חלוקה בת <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6e1f880981346a604257ebcacdef24c0aca2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}"></span> קטעים הוא לפחות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {b-a}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {b-a}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ef83d8500f7eb8232c110916b3a8a2ee24a48e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.69ex; height:3.509ex;" alt="{\textstyle {\frac {b-a}{n}}}"></span>. מכאן שכדי לקרב את קוטר החלוקה לאפס, יש להגדיל את מספר הנקודות באופן שאינו חסום. </p><p>חלוקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msup> <mi>π</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae135820a86db735299ab8f332dc24fa0fe2eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.343ex;" alt="{\textstyle \pi '}"></span> היא <b>עידון</b> של חלוקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27d9f6a29fcb6e9c0c5d94804f9ee4b271e621b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\textstyle \pi }"></span>, אם החלוקה הראשונה כוללת את כל הנקודות המופיעות בחלוקה השנייה. לדוגמה, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a=x_{0}<x_{1}<x_{2}<x_{3}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a=x_{0}<x_{1}<x_{2}<x_{3}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d5f87697dcd222104739b8b9225ede89fda75bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.255ex; height:2.509ex;" alt="{\textstyle a=x_{0}<x_{1}<x_{2}<x_{3}=b}"></span> מעדנת את <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a=x_{0}<x_{2}<x_{3}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a=x_{0}<x_{2}<x_{3}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f0d8af8962f016ccb975da2c62f356ad408bb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.773ex; height:2.509ex;" alt="{\textstyle a=x_{0}<x_{2}<x_{3}=b}"></span>. ברור שבמקרה כזה, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda (\pi ')\leq \lambda (\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda (\pi ')\leq \lambda (\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2681c550fc595f660d37479f8c705ccb8c42109e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.778ex; height:2.843ex;" alt="{\textstyle \lambda (\pi ')\leq \lambda (\pi )}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="הגדרת_האינטגרל_המסוים_באמצעות_סכומי_רימן"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.AA_.D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.94.D7.9E.D7.A1.D7.95.D7.99.D7.9D_.D7.91.D7.90.D7.9E.D7.A6.D7.A2.D7.95.D7.AA_.D7.A1.D7.9B.D7.95.D7.9E.D7.99_.D7.A8.D7.99.D7.9E.D7.9F"></span>הגדרת האינטגרל המסוים באמצעות סכומי רימן</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=4" title="עריכת קוד המקור של הפרק: הגדרת האינטגרל המסוים באמצעות סכומי רימן"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=4" title="עריכת פסקה: "הגדרת האינטגרל המסוים באמצעות סכומי רימן"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:Riemann_sum_convergence.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/300px-Riemann_sum_convergence.png" decoding="async" width="300" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/450px-Riemann_sum_convergence.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Riemann_sum_convergence.png/600px-Riemann_sum_convergence.png 2x" data-file-width="1260" data-file-height="1260" /></a><figcaption>התכנסות סכומי רימן לאינטגרל: בכחול מתוארת חלוקה שבה בוחרים בכל תת־קטע את הנקודה הימנית; בצהוב – הנקודה השמאלית; בירוק – נקודת המקסימום בתת־קטע; וּבאדום – נקודת המינימום. הגרף שבמרכז מתאר את התנהגות ארבעת הקירובים כאשר מספר המלבנים גדל. מכיוון שהפונקציה אינטגרבילית, כל הקירובים שואפים לאותו ערך.</figcaption></figure> <p>בחלוקה נתונה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d99c2af530a82249e7bb239caf9bc5193ee766bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:35.506ex; height:2.509ex;" alt="{\textstyle a=x_{0}<x_{1}<\dotsb <x_{n-1}<x_{n}=b}"></span>, אפשר לבחור נקודה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \xi _{i}\in [x_{i-1},x_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \xi _{i}\in [x_{i-1},x_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08b559f0a68116197a305b60b22c469c841307f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.346ex; height:2.843ex;" alt="{\textstyle \xi _{i}\in [x_{i-1},x_{i}]}"></span> מכל תת־קטע. חלוקה כזו, יחד עם הנקודות שנבחרו מהתת־קטעים, נקראת <b>חלוקה מסומנת</b>. לחלוקה כזו אפשר להגדיר את <b>סכום רימן</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (f,\pi )=\sum _{i=1}^{n}{f(\xi _{i})\cdot |x_{i}-x_{i-1}|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (f,\pi )=\sum _{i=1}^{n}{f(\xi _{i})\cdot |x_{i}-x_{i-1}|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b185191fba01a07dca9cb25704b63a7da13eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.702ex; height:6.843ex;" alt="{\displaystyle \sigma (f,\pi )=\sum _{i=1}^{n}{f(\xi _{i})\cdot |x_{i}-x_{i-1}|}}"></span>. זהו השטח הכולל של המלבנים שבסיסם הוא הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [x_{i-1},x_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [x_{i-1},x_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad785afcc8d0dab1d4a679d9d68c2bda3fedc04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.687ex; height:2.843ex;" alt="{\textstyle [x_{i-1},x_{i}]}"></span> וגובהם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\xi _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(\xi _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aaa238fb251c819fbc00e4685cd952788d1affb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:2.843ex;" alt="{\textstyle f(\xi _{i})}"></span> (עבור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle i=1,\dotsc ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle i=1,\dotsc ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ce0bbd2b3f48334029bb8a9829086945e6e5c6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.636ex; height:2.509ex;" alt="{\textstyle i=1,\dotsc ,n}"></span>), כאשר השטח הוא "שטח מכוון" (העשוי להיות חיובי או שלילי, בהתאם ל<a href="/wiki/%D7%A1%D7%99%D7%9E%D7%9F_(%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" class="mw-redirect" title="סימן (אריתמטיקה)">סימן</a> של הפונקציה בנקודה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \xi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \xi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/025cf1bcc9193425d5a296f2c1433419bf8ce102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.818ex; height:2.509ex;" alt="{\textstyle \xi _{i}}"></span>). כל סכום רימן מהווה קירוב לשטח שמתחת לגרף הפונקציה, בקטע המדובר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span>. </p><p>פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle f(x)}"></span>, המוגדרת בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span>, היא <b>אינטגרבילית לפי רימן</b>, אם לכל בחירה של סדרת חלוקות מסומנות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \pi _{1},\pi _{2},\dotsc }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \pi _{1},\pi _{2},\dotsc }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b6f907ef69f7a4fbd21fc3b40ae823fbead7e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.55ex; height:2.009ex;" alt="{\textstyle \pi _{1},\pi _{2},\dotsc }"></span> בעלות גדלים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \lambda (\pi _{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>λ</mi> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \lambda (\pi _{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efbd8602c48bf86d64c791204c06b66aabc8ed3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.165ex; height:2.843ex;" alt="{\textstyle \lambda (\pi _{m})}"></span> ה<a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_%D7%A9%D7%9C_%D7%A1%D7%93%D7%A8%D7%94" title="גבול של סדרה">שואפים</a> לאפס, הגבול <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6810e30ec459e38d6359417a19a9c9624639960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.568ex; height:3.843ex;" alt="{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}"></span> קיים (היינו, הסדרה מתכנסת). במקרה כזה, כל הגבולות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6810e30ec459e38d6359417a19a9c9624639960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.568ex; height:3.843ex;" alt="{\displaystyle \lim _{m\rightarrow \infty }\sigma (f,\pi _{m})}"></span> שווים זה לזה<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>, והאינטגרל המסוים מוגדר כערך (המשותף) של כל הגבולות; ערך זה הוא השטח שבין גרף הפונקציה לציר ה־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span>. אם לא כל הגבולות האלה קיימים, הפונקציה אינה אינטגרבילית לפי רימן. </p><p>את ערך האינטגרל מסמנים כך: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac02adeed584466d53dee65f3228ad66939eb58b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.139ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx}"></span>. הביטוי שבתוך האינטגרל, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0a982c6635ab3b98d9e12d5f5a8533359bcb38a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\textstyle f(x)}"></span>, נקרא <b>אינטגרנד</b>. המשתנה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d951e0f3b54b6a3d73bb9a0a005749046cbce781" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}"></span> בביטוי זה הוא "משתנה האינטגרציה", והוא קשור לתחום האינטגרציה – המשמעות היא שהנקודה השמאלית בתחום היא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e0fe729bde9223f1b509db0471471c21261b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\textstyle x=a}"></span> והנקודה הימנית היא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66b5e36a6a98d41ca49d85cb317787310a0c797d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\textstyle x=b}"></span>. אם מחליפים את משתנה האינטגרציה, כדי לשמור על ערך האינטגרל יש לשנות את תחום האינטגרציה בהתאמה. לדוגמה, אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle t=kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle t=kx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b81921078120253ba1f21ef9ea652dbff55ec4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.479ex; height:2.176ex;" alt="{\textstyle t=kx}"></span>, אזי מתקיים: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{ka}^{kb}{\tfrac {1}{k}}f({\tfrac {t}{k}})\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mstyle> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>t</mi> <mi>k</mi> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{ka}^{kb}{\tfrac {1}{k}}f({\tfrac {t}{k}})\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aafadaaa551bf944d5196442f43e9c233dfa8bfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.799ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,dx=\int _{ka}^{kb}{\tfrac {1}{k}}f({\tfrac {t}{k}})\,dt}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="הגדרה_באמצעות_סכומי_דארבו"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94_.D7.91.D7.90.D7.9E.D7.A6.D7.A2.D7.95.D7.AA_.D7.A1.D7.9B.D7.95.D7.9E.D7.99_.D7.93.D7.90.D7.A8.D7.91.D7.95"></span>הגדרה באמצעות סכומי דארבו</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=5" title="עריכת קוד המקור של הפרק: הגדרה באמצעות סכומי דארבו"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=5" title="עריכת פסקה: "הגדרה באמצעות סכומי דארבו"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:RightRiemann2.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/170px-RightRiemann2.PNG" decoding="async" width="170" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/255px-RightRiemann2.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/340px-RightRiemann2.PNG 2x" data-file-width="640" data-file-height="480" /></a><figcaption>חישוב הסכום העליון של פונקציה</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:LeftRiemann2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/LeftRiemann2.png/170px-LeftRiemann2.png" decoding="async" width="170" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/LeftRiemann2.png/255px-LeftRiemann2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/LeftRiemann2.png/340px-LeftRiemann2.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>חישוב הסכום התחתון של פונקציה</figcaption></figure> <p>נניח ש־ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%97%D7%A1%D7%95%D7%9E%D7%94" title="פונקציה חסומה">חסומה</a> ב<a href="/wiki/%D7%A7%D7%98%D7%A2" class="mw-redirect" title="קטע">קטע</a> ה<a href="/wiki/%D7%A7%D7%98%D7%A2_%D7%A1%D7%92%D7%95%D7%A8" class="mw-redirect" title="קטע סגור">סגור</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span>. לכל חלוקה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, אפשר לחשב בנפרד את השטח שהחלוקה מאתרת מתחת לגרף, ואת השטח שהחלוקה מאתרת מעל לגרף. לצורך כך נסמן בכל תת-קטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x_{i-1},x_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{i-1},x_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09cb12a889d47020c8ce7046a2eb60785e00c0b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.687ex; height:2.843ex;" alt="{\displaystyle [x_{i-1},x_{i}]}"></span> של החלוקה, את ה<a href="/wiki/%D7%97%D7%A1%D7%9D_%D7%A2%D7%9C%D7%99%D7%95%D7%9F" class="mw-redirect" title="חסם עליון">חסם העליון</a> של הפונקציה ב־ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"></span>, ואת החסם התחתון ב־ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"></span> (אם לפונקציה יש בתת-הקטע הזה ערך מינימלי או מקסימלי, אלו יהיו הערכים של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ec8e804f69706d3f5ad235f4f983220c8df7c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.009ex;" alt="{\displaystyle m_{i}}"></span> ושל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eda8fd06f1cd5de22ed07385a0f8aa19773b2de9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.054ex; height:2.509ex;" alt="{\displaystyle M_{i}}"></span> בהתאמה). באופן הזה, מובטח ש־ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}\leq f(t)\leq M_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}\leq f(t)\leq M_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e823152dfbd781a0f32fe7396e597aa67af04e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.018ex; height:2.843ex;" alt="{\displaystyle m_{i}\leq f(t)\leq M_{i}}"></span> לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> בתת-הקטע. משום כך סביר לקבוע ששטחו של התחום המישורי המוגבל על ידי ציר ה־x, גרף הפונקציה, והישרים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{i-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{i-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f5576689f2ffc87639c81b53496ba8793977df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.658ex; height:2.009ex;" alt="{\displaystyle x=x_{i-1}}"></span> ו־ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b06a17bffa1d6a69d2a0c908917d20386622b6f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.558ex; height:2.009ex;" alt="{\displaystyle x=x_{i}}"></span>, גדול או שווה לשטח המלבן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{i}|x_{i}-x_{i-1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i}|x_{i}-x_{i-1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ceb8953e75493389715c8e5d191fd3dc79667e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.333ex; height:2.843ex;" alt="{\displaystyle m_{i}|x_{i}-x_{i-1}|}"></span>, וקטן (או שווה) לשטח המלבן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{i}|x_{i}-x_{i-1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{i}|x_{i}-x_{i-1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1564e9c70983f042457324518c9913d1a823b123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.547ex; height:2.843ex;" alt="{\displaystyle M_{i}|x_{i}-x_{i-1}|}"></span>. </p><p>אם נסכם את כל המלבנים, הסכום <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {S}}(\pi )=\sum _{i=1}^{n}m_{i}|x_{i}-x_{i-1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>S</mi> <mo>_</mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {S}}(\pi )=\sum _{i=1}^{n}m_{i}|x_{i}-x_{i-1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c0e5fbdcfcff8b2664111c147e6de80c4613123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.048ex; width:24.864ex; height:6.843ex;" alt="{\displaystyle {\underline {S}}(\pi )=\sum _{i=1}^{n}m_{i}|x_{i}-x_{i-1}|}"></span> נקרא <b>הסכום התחתון</b> של החלוקה, ואילו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}(\pi )=\sum _{i=1}^{n}M_{i}|x_{i}-x_{i-1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}(\pi )=\sum _{i=1}^{n}M_{i}|x_{i}-x_{i-1}|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2acfbbd4f2396ab2c3bbf68532401725bb5ba84a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.191ex; height:6.843ex;" alt="{\displaystyle {\overline {S}}(\pi )=\sum _{i=1}^{n}M_{i}|x_{i}-x_{i-1}|}"></span> הוא <b>הסכום העליון</b> שלה. קל להוכיח שאם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi '}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>π</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi '}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96621a3b005d05c898c2f1fb594134fdf35e476d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.509ex;" alt="{\displaystyle \pi '}"></span> מהווה עידון של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>, אז <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {S}}(\pi )\leq {\underline {S}}(\pi ')\leq {\overline {S}}(\pi ')\leq {\overline {S}}(\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>S</mi> <mo>_</mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>S</mi> <mo>_</mo> </munder> </mrow> <mo stretchy="false">(</mo> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <msup> <mi>π</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {S}}(\pi )\leq {\underline {S}}(\pi ')\leq {\overline {S}}(\pi ')\leq {\overline {S}}(\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2a4421925a812372192dbc1f7e6035493575622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.048ex; margin-bottom: -0.755ex; width:29.609ex; height:4.009ex;" alt="{\displaystyle {\underline {S}}(\pi )\leq {\underline {S}}(\pi ')\leq {\overline {S}}(\pi ')\leq {\overline {S}}(\pi )}"></span>, ולכן, כאשר מעדנים את החלוקה, המרחק בין הסכום העליון לתחתון מצטמצם. </p><p>החסם התחתון של כל הסכומים העליונים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}(\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}(\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4aec3db6b04157418dd31752785d303073ecab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.804ex; height:3.509ex;" alt="{\displaystyle {\overline {S}}(\pi )}"></span>, עבור כל החלוקות האפשריות, הוא <b>האינטגרל העליון</b>. החסם העליון של כל הסכומים התחתונים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {S}}(\pi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>S</mi> <mo>_</mo> </munder> </mrow> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {S}}(\pi )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b9a22d4635e4cd09300ffbdf07cb98ca228d460" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.048ex; margin-bottom: -0.755ex; width:4.69ex; height:3.343ex;" alt="{\displaystyle {\underline {S}}(\pi )}"></span> הוא <b>האינטגרל התחתון</b>. הפונקציה <b>אינטגרבילית לפי <a href="/wiki/%D7%96%27%D7%90%D7%9F_%D7%92%D7%A1%D7%98%D7%95%D7%9F_%D7%93%D7%90%D7%A8%D7%91%D7%95" title="ז'אן גסטון דארבו">דארבו</a></b>, אם שני ערכים אלו שווים זה לזה (פירושו של השוויון הוא שקיימת סדרה של חלוקות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3144ebd68015a67d92ab797a63d232c65ead26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.544ex; height:2.009ex;" alt="{\displaystyle \pi _{n}}"></span> המעדנות זו את זו, כך שההפרש <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {S}}(\pi _{n})-{\underline {S}}(\pi _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>S</mi> <mo>_</mo> </munder> </mrow> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {S}}(\pi _{n})-{\underline {S}}(\pi _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb6ac54b8448f24e4642103a3a80e49a8ecb4ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:14.709ex; height:4.009ex;" alt="{\displaystyle {\overline {S}}(\pi _{n})-{\underline {S}}(\pi _{n})}"></span> שואף לאפס). </p><p>אפשר להוכיח שהגדרת אינטגרביליות של פונקציה חסומה באמצעות סכומי רימן שקולה להגדרה באמצעות סכומי דארבו, ושהאינטגרל המתקבל בשני המקרים שווה. ההגדרה שנתנה לעיל מתאימה לפונקציות חסומות, ולחישוב מעל קטע סגור. עם זאת, אפשר להרחיב את ההגדרה גם למקרים כלליים יותר – ראו <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%90_%D7%90%D7%9E%D7%99%D7%AA%D7%99" title="אינטגרל לא אמיתי">אינטגרל לא אמיתי</a>. </p> <div class="mw-heading mw-heading3"><h3 id="מרחב_הפונקציות_האינטגרביליות"><span id=".D7.9E.D7.A8.D7.97.D7.91_.D7.94.D7.A4.D7.95.D7.A0.D7.A7.D7.A6.D7.99.D7.95.D7.AA_.D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.91.D7.99.D7.9C.D7.99.D7.95.D7.AA"></span>מרחב הפונקציות האינטגרביליות</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=6" title="עריכת קוד המקור של הפרק: מרחב הפונקציות האינטגרביליות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=6" title="עריכת פסקה: "מרחב הפונקציות האינטגרביליות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>הסכום של פונקציות אינטגרביליות (לפי רימן) והכפולה של פונקציה אינטגרבילית בסקלר נותנים פונקציות אינטגרביליות; לכן אוסף הפונקציות האינטגרביליות מעל קטע קבוע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span> מהווה <a href="/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99" title="מרחב וקטורי">מרחב וקטורי</a> מעל <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">שדה המספרים הממשיים</a>, ואף מוגדרת בו <a href="/wiki/%D7%9E%D7%9B%D7%A4%D7%9C%D7%94_%D7%A4%D7%A0%D7%99%D7%9E%D7%99%D7%AA" class="mw-redirect" title="מכפלה פנימית">מכפלה פנימית</a>. </p><p><b>משפט לבג</b> מאפיין אינטגרביליות באופן הבא: פונקציה חסומה היא אינטגרבילית במובן רימן, <a href="/wiki/%D7%90%D7%9D_%D7%95%D7%A8%D7%A7_%D7%90%D7%9D" title="אם ורק אם">אם ורק אם</a> קבוצת <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" class="mw-redirect" title="רציפות">נקודות אי־הרציפות</a> שלה היא בעלת <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%9E%D7%9E%D7%99%D7%93%D7%94_%D7%90%D7%A4%D7%A1" title="קבוצה ממידה אפס">מידה אפס</a>. </p><p>לדוגמה, כל פונקציה רציפה וכל <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%95%D7%A0%D7%95%D7%98%D7%95%D7%A0%D7%99%D7%AA" title="פונקציה מונוטונית">פונקציה מונוטונית</a> בקטע סגור, היא אינטגרבילית. גם <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%A8%D7%99%D7%9E%D7%9F" title="פונקציית רימן">פונקציית רימן</a> אינטגרבילית (והאינטגרל שלה הוא אפס). לעומת זאת, <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%93%D7%99%D7%A8%D7%99%D7%9B%D7%9C%D7%94" title="פונקציית דיריכלה">פונקציית דיריכלה</a> אינה אינטגרבילית לפי רימן. </p><p>ההתאמה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto \int _{a}^{b}\!\!f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto \int _{a}^{b}\!\!f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79341d498e099d32428d55cd8cf373a42bdc730e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.87ex; height:6.343ex;" alt="{\displaystyle f\mapsto \int _{a}^{b}\!\!f(x)dx}"></span> היא <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99" class="mw-redirect" title="פונקציונל ליניארי">פונקציונל ליניארי</a> המוגדר על המרחב הזה, משום שהאינטגרל של פונקציות מקיים את התכונה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}{[c_{1}f(x)+c_{2}g(x)]dx}=c_{1}\int _{a}^{b}{f(x)dx}+c_{2}\int _{a}^{b}{g(x)dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>d</mi> <mi>x</mi> </mrow> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}{[c_{1}f(x)+c_{2}g(x)]dx}=c_{1}\int _{a}^{b}{f(x)dx}+c_{2}\int _{a}^{b}{g(x)dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdec2e902f1971087d9365d9e6e8fb145caaeb47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:55.44ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}{[c_{1}f(x)+c_{2}g(x)]dx}=c_{1}\int _{a}^{b}{f(x)dx}+c_{2}\int _{a}^{b}{g(x)dx}}"></span>. האינטגרל <b>מונוטוני</b>, במובן הבא: אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}"></span> אינטגרביליות בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c780cbaafb5b1d4a6912aa65d2b0b1982097108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\textstyle [a,b]}"></span>, ולכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.725ex; height:2.843ex;" alt="{\displaystyle x\in [a,b]}"></span> מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\geq g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)\geq g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3686eceede316fd569eda1d2743cdfb7a1bd41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle f(x)\geq g(x)}"></span>, אז <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}{f(x)dx}\geq \int _{a}^{b}{g(x)dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> <mo>≥</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}{f(x)dx}\geq \int _{a}^{b}{g(x)dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2719a4bb7e11e90fbe6858ac8049ffa7aa2ff01e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.44ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}{f(x)dx}\geq \int _{a}^{b}{g(x)dx}}"></span> ואם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)>g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>></mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)>g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6155f59752178439a14291d5cd00d2dbe0ed07ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.771ex; height:2.843ex;" alt="{\displaystyle f(x)>g(x)}"></span> לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle x\in [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b494d0964a68e4a9a124ee1b6e512843a1b14a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.725ex; height:2.843ex;" alt="{\textstyle x\in [a,b]}"></span> אז מתקיים אי השוויון החזק <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}{f(x)dx}>\int _{a}^{b}{g(x)dx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> <mo>></mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}{f(x)dx}>\int _{a}^{b}{g(x)dx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f24e9fcfb617b2ea35824389ba8efe7b3e2bbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.44ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}{f(x)dx}>\int _{a}^{b}{g(x)dx}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="חישוב_האינטגרל_המסוים"><span id=".D7.97.D7.99.D7.A9.D7.95.D7.91_.D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.94.D7.9E.D7.A1.D7.95.D7.99.D7.9D"></span>חישוב האינטגרל המסוים</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=7" title="עריכת קוד המקור של הפרק: חישוב האינטגרל המסוים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=7" title="עריכת פסקה: "חישוב האינטגרל המסוים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%D7%94%D7%A0%D7%95%D7%A1%D7%97%D7%94_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99%D7%AA_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" class="mw-redirect" title="הנוסחה היסודית של החשבון האינפיניטסימלי">הנוסחה היסודית של החשבון האינפיניטסימלי</a> מחשבת את האינטגרל המסוים, אם ידוע האינטגרל הלא־מסוים (ראו להלן). במקרים אחרים יש להפעיל <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%95%D7%AA_%D7%9C%D7%97%D7%99%D7%A9%D7%95%D7%91_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99%D7%9D_%D7%9E%D7%A1%D7%95%D7%99%D7%9E%D7%99%D7%9D" title="שיטות אנליטיות לחישוב אינטגרלים מסוימים">שיטות אנליטיות</a> מיוחדות, או <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%A0%D7%95%D7%9E%D7%A8%D7%99%D7%95%D7%AA_%D7%9C%D7%97%D7%99%D7%A9%D7%95%D7%91_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99%D7%9D_%D7%9E%D7%A1%D7%95%D7%99%D7%9E%D7%99%D7%9D" title="שיטות נומריות לחישוב אינטגרלים מסוימים">שיטות נומריות</a>. </p> <div class="mw-heading mw-heading2"><h2 id="האינטגרל_הלא_מסוים"><span id=".D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.94.D7.9C.D7.90_.D7.9E.D7.A1.D7.95.D7.99.D7.9D"></span>האינטגרל הלא מסוים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=8" title="עריכת קוד המקור של הפרק: האינטגרל הלא מסוים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=8" title="עריכת פסקה: "האינטגרל הלא מסוים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="הגדרה"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94"></span>הגדרה</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=9" title="עריכת קוד המקור של הפרק: הגדרה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=9" title="עריכת פסקה: "הגדרה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b04e6ce1205822c3f29f3f1448160fb97cfd8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.267ex; height:2.843ex;" alt="{\displaystyle F\left(x\right)}"></span> נקראת <b><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%93%D7%95%D7%9E%D7%94" class="mw-redirect" title="פונקציה קדומה">פונקציה קדומה</a></b> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653b89efce2f12f2c8bb8a5536ac569fe73e8271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f\left(x\right)}"></span> בקטע כלשהו, אם לכל נקודה x בקטע מתקיים:<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'\left(x\right)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'\left(x\right)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d264a38cb1f05179f273b3034e564c8139029b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.542ex; height:3.009ex;" alt="{\displaystyle F'\left(x\right)=f(x)}"></span>, כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653b89efce2f12f2c8bb8a5536ac569fe73e8271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.805ex; height:2.843ex;" alt="{\displaystyle f\left(x\right)}"></span> היא ה<a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\left(x\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\left(x\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35b04e6ce1205822c3f29f3f1448160fb97cfd8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.267ex; height:2.843ex;" alt="{\displaystyle F\left(x\right)}"></span> בקטע. </p><p><b>האינטגרל הלא מסוים</b> של פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> המוגדרת בשדה F כלשהו מוגדר לרוב בתור אוסף הפונקציות הקדומות שלו. בסימון: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \!\!f(x)\,\mathrm {d} x=F(x)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mspace width="negativethinmathspace" /> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \!\!f(x)\,\mathrm {d} x=F(x)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b660eecd27b46449a3b673fa4945454b412b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.818ex; height:5.676ex;" alt="{\displaystyle \int \!\!f(x)\,\mathrm {d} x=F(x)+C}"></span>, כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> היא פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> ו־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\in \mathbb {F} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\in \mathbb {F} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/990ba986be5e2059a8229771d873b03b1567d041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C\in \mathbb {F} }"></span> הוא קבוע שרירותי. </p><p>ניתן להצדיק את הסימון בכך שכל הפונקציות הקדומות של פונקציה ניתנות לכתיבה בתור קבוע ועוד פונקציה קדומה כלשהי. מצד אחד, אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(x)=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x)=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5457591f5410f4bfe3b9c9fa2e50ae665fa2822c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.154ex; height:3.009ex;" alt="{\displaystyle F'(x)=f(x)}"></span> אז ברור כי גם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(F(x)+C\right)'=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(F(x)+C\right)'=f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6aa72c337f4aa1e677983dd930c377910f6194" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.497ex; height:3.176ex;" alt="{\displaystyle \left(F(x)+C\right)'=f(x)}"></span> כי נגזרת של קבוע היא 0. מצד שני, אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x),G(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x),G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b7ccc985958aeb0f4c03f3ff3745a4342911903" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.879ex; height:2.843ex;" alt="{\displaystyle F(x),G(x)}"></span> פונקציות קדומות של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> אז מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(F(x)-G(x)\right)'=f(x)-f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>(</mo> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(F(x)-G(x)\right)'=f(x)-f(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f506258ccd7546d8cf661d970d92114f46cc5812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.215ex; height:3.176ex;" alt="{\displaystyle \left(F(x)-G(x)\right)'=f(x)-f(x)=0}"></span>, כלומר הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)-G(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)-G(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c70072b3b9a2025e8368d24c64a5df21ff2034" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.686ex; height:2.843ex;" alt="{\displaystyle F(x)-G(x)}"></span> קבועה, כלומר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)-G(x)=C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)-G(x)=C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f04a2d436e31d4671093a78c07358049431268f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.551ex; height:2.843ex;" alt="{\displaystyle F(x)-G(x)=C}"></span>, כנדרש. </p><p>בהינתן פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> אינטגרבילית בקטע הסגור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> ניתן להגדיר פונקציה באופן הבא: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x\in [a,b]:\ \ \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> <mo>:</mo> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in [a,b]:\ \ \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc75b7da516fad8e0975566b9f1d3aea9e56c847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.146ex; height:5.843ex;" alt="{\displaystyle \forall x\in [a,b]:\ \ \ F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span></dd></dl> <p>ערכה של פונקציה זו בכל נקודה הוא ערך האינטגרל המסוים של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> בין נקודה זו לנקודת מוצא כלשהי. פונקציה זו היא תמיד <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%94" class="mw-redirect" title="רציפה">רציפה</a>, אך אינה בהכרח <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">גזירה</a> ולכן אינה בהכרח פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span>. עבור <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%9E%D7%93%D7%A8%D7%92%D7%94" title="פונקציית מדרגה">פונקציית מדרגה</a>, למשל, לא יהיה אינטגרל זה גזיר, שכן פונקציית מדרגה אינה מקיימת את תכונת הנגזרת הבאה לידי ביטוי ב<a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%93%D7%90%D7%A8%D7%91%D7%95" title="משפט דארבו">משפט דארבו</a>. </p><p><a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a> אומר ש <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> גזירה בכל מקום בו <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> רציפה. כלומר: אם <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> רציפה ב <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f21d0e31751534cd6584264ecf864a6aa792cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}"></span> אזי מתקיים ש <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F'(x_{0})=f(x_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>F</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F'(x_{0})=f(x_{0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bc24e9d2287452424fe2c8181a07da65bdc8cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.263ex; height:3.009ex;" alt="{\displaystyle F'(x_{0})=f(x_{0})}"></span>. כלומר, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> היא פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> באופן כללי, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82805d469cdfa7856c11d6ee756acd1dc7174" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle F(x)}"></span> לא חייבת להיות גזירה בכל מקום. </p><p>על כן, המשפט היסודי קושר בין האינטגרל המסוים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24411c18e9fb6a78762e27868d58cf4fbf0a883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.449ex; height:5.843ex;" alt="{\displaystyle F(x)=\int _{a}^{x}f(t)\,\mathrm {d} t}"></span> ובין האינטגרל הלא מסוים של הפונקציה, וממנו נגזרת הנוסחה היסודית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=F(b)-F(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=F(b)-F(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e272c5ec268675124f4f31b6c551aebe450fa6ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.482ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=F(b)-F(a)}"></span> המאפשרת לחשב אינטגרל מסוים באמצעות שימוש בפונקציה קדומה. </p> <div class="mw-heading mw-heading3"><h3 id="דוגמה"><span id=".D7.93.D7.95.D7.92.D7.9E.D7.94"></span>דוגמה</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=10" title="עריכת קוד המקור של הפרק: דוגמה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=10" title="עריכת פסקה: "דוגמה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>ידוע כי הנגזרת של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> היא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50b849d3a7cd902f0ae3fa6ad6d1cad49987c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 2x}"></span>‏. על כן כל פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50b849d3a7cd902f0ae3fa6ad6d1cad49987c39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 2x}"></span> נבדלת מ-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"></span> בקבוע, ונכתוב: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int 2x\,dx=x^{2}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mn>2</mn> <mi>x</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int 2x\,dx=x^{2}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e497bbdc8e691a8b939d3a27fa5d2723f75e0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.335ex; height:5.676ex;" alt="{\displaystyle \int 2x\,dx=x^{2}+c}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="מציאת_האינטגרל_הלא_מסוים"><span id=".D7.9E.D7.A6.D7.99.D7.90.D7.AA_.D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.94.D7.9C.D7.90_.D7.9E.D7.A1.D7.95.D7.99.D7.9D"></span>מציאת האינטגרל הלא מסוים</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=11" title="עריכת קוד המקור של הפרק: מציאת האינטגרל הלא מסוים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=11" title="עריכת פסקה: "מציאת האינטגרל הלא מסוים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>בניגוד לפעולת הגזירה, שהיא טכנית בעיקרה ומבוססת על כמה כללים ברורים היטב, אין "מתכון" בטוח למציאת אינטגרל לא מסוים של פונקציה. באמצעות נוסחאות הגזירה ניתן למצוא מיידית אינטגרלים לפונקציות האלמנטריות הבסיסיות, ועל מנת לבצע אינטגרציה לפונקציות מסובכות יותר ישנן <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" title="שיטות אינטגרציה">שיטות אינטגרציה</a> (<a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94_%D7%91%D7%90%D7%9E%D7%A6%D7%A2%D7%95%D7%AA_%D7%94%D7%97%D7%9C%D7%A4%D7%AA_%D7%9E%D7%A9%D7%AA%D7%A0%D7%99%D7%9D" title="אינטגרציה באמצעות החלפת משתנים">החלפת משתנים</a>, <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94_%D7%91%D7%97%D7%9C%D7%A7%D7%99%D7%9D" title="אינטגרציה בחלקים">אינטגרציה בחלקים</a> ועוד) שמאפשרות לפשט את הפונקציה ולהפוך אותה לפונקציה אחרת, שעבורה קל יותר למצוא את האינטגרל. </p><p>גם אם לא ניתן לבטא את האינטגרל הלא מסוים באמצעות <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%90%D7%A0%D7%9C%D7%99%D7%98%D7%99%D7%AA" title="פונקציה אנליטית">פונקציה אנליטית</a>, אין זה אומר שהאינטגרל המסוים אינו קיים. בהרבה מקרים (למשל ב<a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%95%D7%AA_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%95%D7%AA" class="mw-redirect" title="משוואות דיפרנציאליות">משוואות דיפרנציאליות</a>) ביטויים מהצורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(x)=y_{0}+\int _{x_{0}}^{x}f(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(x)=y_{0}+\int _{x_{0}}^{x}f(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc0d71990c932fa3515f73f9557751839f09e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.897ex; height:6.176ex;" alt="{\displaystyle y(x)=y_{0}+\int _{x_{0}}^{x}f(t)\,\mathrm {d} t}"></span> נחשבים לפתרון קביל. </p> <div class="mw-heading mw-heading4"><h4 id="הערות"><span id=".D7.94.D7.A2.D7.A8.D7.95.D7.AA"></span>הערות</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=12" title="עריכת קוד המקור של הפרק: הערות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=12" title="עריכת פסקה: "הערות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>למעשה, סכום רימן הוא חלוקה של הקטע ל<a href="/wiki/%D7%9E%D7%9C%D7%91%D7%9F" title="מלבן">מלבנים</a> צרים שגובהם הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\xi _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\xi _{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36ab9ff2036e242bc547385a26136dda809aafda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:2.843ex;" alt="{\displaystyle f(\xi _{i})}"></span>, סיכום שטחיהם ומעבר ל<a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="גבול (מתמטיקה)">גבול</a> כאשר פרמטר החלוקה שואף לאפס. ב<a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%A0%D7%95%D7%9E%D7%A8%D7%99%D7%AA" title="אנליזה נומרית">אנליזה נומרית</a> יש חשיבות גדולה לבחירת נקודות הביניים כדי לקבל התכנסות מהירה של הקירוב הנומרי לערך המדויק (שלרוב אינו ניתן לחישוב). </p> <div class="mw-heading mw-heading2"><h2 id="הכללות_של_אינטגרל_רימן"><span id=".D7.94.D7.9B.D7.9C.D7.9C.D7.95.D7.AA_.D7.A9.D7.9C_.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.A8.D7.99.D7.9E.D7.9F"></span>הכללות של אינטגרל רימן</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=13" title="עריכת קוד המקור של הפרק: הכללות של אינטגרל רימן"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=13" title="עריכת פסקה: "הכללות של אינטגרל רימן"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="אינטגרל_לבג"><span id=".D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.9C.D7.91.D7.92"></span>אינטגרל לבג</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=14" title="עריכת קוד המקור של הפרק: אינטגרל לבג"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=14" title="עריכת פסקה: "אינטגרל לבג"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/25px-Postscript-viewer-blue.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/38px-Postscript-viewer-blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/50px-Postscript-viewer-blue.svg.png 2x" data-file-width="60" data-file-height="60" /></span></span> ערך מורחב – <b><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%91%D7%92" title="אינטגרל לבג">אינטגרל לבג</a></b><br /></dd></dl> <p>אינטגרל לבג הוא הכללה של אינטגרל רימן ל<a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%93%D7%99%D7%93%D7%94" title="פונקציה מדידה">פונקציות מדידות</a> שפותחה על ידי ה<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%90%D7%99" title="מתמטיקאי">מתמטיקאי</a> <a href="/wiki/%D7%90%D7%A0%D7%A8%D7%99_%D7%9C%D7%91%D7%92" title="אנרי לבג">אנרי לבג</a> במסגרת מחקרו על <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a>. אינטגרל לבג מתבסס על <a href="/wiki/%D7%9E%D7%99%D7%93%D7%AA_%D7%9C%D7%91%D7%92" title="מידת לבג">מידת לבג</a> המוגדרת מעל <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">הישר הממשי</a> והוא מזדהה עם אינטגרל רימן לכל פונקציה חסומה שהיא אינטגרבילית במובן רימן. </p><p>בהיות האינטגרל של לבג הכללה של אינטגרל רימן, מאפשר מושג זה לחשב אינטגרל לפונקציות שאינן אינטגרביליות במובן רימן. הרעיון באינטגרל לבג הוא לחשב את השטח לפי התמונה של הפונקציה ולא לפי התחום שלה. היתרון בגישה זו היא שלרוב התמונה של הפונקציה פשוטה בהרבה ו"פתולוגית" הרבה פחות מתחום הגדרתה. לכן, מחלקת הפונקציות שהן אינטגרביליות במובן לבג רחבה יותר ממחלקת הפונקציות האינטגרביליות רימן. למעשה, גם פונקציות שאינן <a href="/wiki/%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" class="mw-redirect" title="רציפות">רציפות</a> באף מקום יכולות להיות אינטגרביליות לבג (בעוד שאינן אינטגרביליות רימן). אחת הדוגמאות הבסיסיות והיפות לפונקציה כזאת היא <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%99%D7%AA_%D7%93%D7%99%D7%A8%D7%99%D7%9B%D7%9C%D7%94" title="פונקציית דיריכלה">פונקציית דיריכלה</a>. </p> <div class="mw-heading mw-heading3"><h3 id="אינטגרל_רימן־סטילטיס_ואינטגרל_לבג־סטילטיס"><span id=".D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.A8.D7.99.D7.9E.D7.9F.D6.BE.D7.A1.D7.98.D7.99.D7.9C.D7.98.D7.99.D7.A1_.D7.95.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C_.D7.9C.D7.91.D7.92.D6.BE.D7.A1.D7.98.D7.99.D7.9C.D7.98.D7.99.D7.A1"></span>אינטגרל רימן־סטילטיס ואינטגרל לבג־סטילטיס</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=15" title="עריכת קוד המקור של הפרק: אינטגרל רימן־סטילטיס ואינטגרל לבג־סטילטיס"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=15" title="עריכת פסקה: "אינטגרל רימן־סטילטיס ואינטגרל לבג־סטילטיס"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%99%D7%9E%D7%9F-%D7%A1%D7%98%D7%99%D7%9C%D7%98%D7%99%D7%A1&action=edit&redlink=1" class="new" title="אינטגרל רימן-סטילטיס (הדף אינו קיים)">אינטגרל רימן-סטילטיס</a> הוא הכללה אחרת של אינטגרל רימן. </p><p>אינטגרל רימן־סטילטיס של פונקציה ממשית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90e6f33bba8a2148bbdfde754ff18525694e30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.666ex; height:2.509ex;" alt="{\displaystyle \,f}"></span> של משתנה ממשי ביחס לפונקציה ממשית <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bd980ef3755427fdcb153b4b37eb38522a7429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.503ex; height:2.009ex;" alt="{\displaystyle \,g}"></span> מסומן: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f1a9752af0bd98f69e3e1c80d260b26b0a16a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.141ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} g(x)}"></span></dd></dl> <p>ומוגדר להיות הגבול של הביטוי: </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{x_{i}\in P}f(c_{i})(g(x_{i+1})-g(x_{i}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>P</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{x_{i}\in P}f(c_{i})(g(x_{i+1})-g(x_{i}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fecf0b5e4b240bd99d20c064b81b9f178ae930f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.037ex; height:5.843ex;" alt="{\displaystyle \sum _{x_{i}\in P}f(c_{i})(g(x_{i+1})-g(x_{i}))}"></span> </p><p>כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,c_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88d0994a3d2e594384e2db35931bbe01778a7d63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle \,c_{i}}"></span> נמצא ברווח ה־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f624ef21533c20e2a78b1e22f156069a6689c0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.19ex; height:2.176ex;" alt="{\displaystyle \,i}"></span> בחלוקת הקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,[a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,[a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3481ed60d2bbdaaa1c3cbb8fc27f260c21615f89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.942ex; height:2.843ex;" alt="{\displaystyle \,[a,b]}"></span> לקטעים וכאשר אורך הקטע המקסימלי בחלוקה שואף ל־0. </p><p>האינטגרל אינו מוגדר כאשר לשתי הפונקציות <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90e6f33bba8a2148bbdfde754ff18525694e30a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.666ex; height:2.509ex;" alt="{\displaystyle \,f}"></span> ו־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bd980ef3755427fdcb153b4b37eb38522a7429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.503ex; height:2.009ex;" alt="{\displaystyle \,g}"></span> יש נקודת אי־רציפות משותפת. יש הכללה שתגדיר את האינטגרל כאשר בנקודת אי הרציפות המשותפת אחת הפונקציות רציפה מימין והשנייה משמאל. </p><p>הכללה נוספת היא <a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%91%D7%92-%D7%A1%D7%98%D7%99%D7%9C%D7%98%D7%99%D7%A1&action=edit&redlink=1" class="new" title="אינטגרל לבג-סטילטיס (הדף אינו קיים)">אינטגרל לבג-סטילטיס</a>, שהוא הכללה הן של אינטגרל רימן־סטילטיס והן של אינטגרל לבג. שתי ההגדרות, של אינטגרל רימן־סטילטיס ושל אינטגרל לבג־סטילטיס הן הגדרות זהות כאשר הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bd980ef3755427fdcb153b4b37eb38522a7429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.503ex; height:2.009ex;" alt="{\displaystyle \,g}"></span> היא פונקציה מונוטונית עולה, וזהו המקרה בו אינטגרל זה משמש ב<a href="/wiki/%D7%A1%D7%98%D7%98%D7%99%D7%A1%D7%98%D7%99%D7%A7%D7%94" title="סטטיסטיקה">סטטיסטיקה</a> וב<a href="/wiki/%D7%9E%D7%A9%D7%AA%D7%A0%D7%94_%D7%9E%D7%A7%D7%A8%D7%99" title="משתנה מקרי">משתנים מקריים</a> כאשר הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98bd980ef3755427fdcb153b4b37eb38522a7429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.503ex; height:2.009ex;" alt="{\displaystyle \,g}"></span> היא פונקציה ההסתברות (המצטברת). </p> <div class="mw-heading mw-heading2"><h2 id="שימושי_האינטגרל"><span id=".D7.A9.D7.99.D7.9E.D7.95.D7.A9.D7.99_.D7.94.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.9C"></span>שימושי האינטגרל</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=16" title="עריכת קוד המקור של הפרק: שימושי האינטגרל"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=16" title="עריכת פסקה: "שימושי האינטגרל"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>שימוש חשוב של האינטגרל הוא מציאת <a href="/wiki/%D7%A9%D7%98%D7%97" title="שטח">שטח</a>. השטח בין פונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> אי־שלילית (כלומר: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall x:f(x)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x:f(x)\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6144d78c6b6f5eb1a4a5f92435e45234930d065c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.238ex; height:2.843ex;" alt="{\displaystyle \forall x:f(x)\geq 0}"></span> ) בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> ובין ציר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c980562004d96eee429ecea3062e59076825a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.216ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x}"></span>. כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> מקבלת גם ערכים שליליים, האינטגרל מחשב את השטח שכלוא בין גרף הפונקציה לציר x אך מחזיר אותו עם סימן בהתאם למיקום של השטח ביחס לציר ה־x: סימן חיובי אם השטח כלוא מעל ציר ה־x וסימן שלילי אם השטח כלוא מתחתיו. למשל: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-1}^{0}xdx=-1/2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> <mi>x</mi> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{0}xdx=-1/2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1709c44493d6db8bbd5f8af9314b3bd5fc85475f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.282ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{0}xdx=-1/2}"></span>. כך ייתכן למשל, ששטחים יקזזו זה את זה כאשר חלק מהם מעל לציר ה־x וחלק מהם מתחתיו. למשל: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-1}^{+1}x\,\mathrm {d} x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{+1}x\,\mathrm {d} x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7dbed59e2fe8e835ee732861f04e4a6333bf78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.784ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{+1}x\,\mathrm {d} x=0}"></span> בעוד ש-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-1}^{+1}|x|\,\mathrm {d} x=2\int _{0}^{1}x\,\mathrm {d} x=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-1}^{+1}|x|\,\mathrm {d} x=2\int _{0}^{1}x\,\mathrm {d} x=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5cc9eb01128ca13b431a341616e45671552ff8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.97ex; height:6.343ex;" alt="{\displaystyle \int _{-1}^{+1}|x|\,\mathrm {d} x=2\int _{0}^{1}x\,\mathrm {d} x=1}"></span>. </p><p>אורך של גרף הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> שהיא גזירה ברציפות בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}{\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}{\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb533796c23c3d1614a6a3474d7fe873381aa3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.133ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}{\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}"></span>. זאת כמקרה פרטי של הנוסחה לאורך <a href="/wiki/%D7%A2%D7%A7%D7%95%D7%9E%D7%94" title="עקומה">עקומה</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54fa4a5d64e164410e4a18106677bebefe1a1f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\displaystyle \gamma (t)}"></span> גזירה ברציפות ורגולרית: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}\|\gamma '(t)\|\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>γ</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}\|\gamma '(t)\|\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f44122545f1efad82ebfefa8c4d40427c8738cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.247ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}\|\gamma '(t)\|\,\mathrm {d} t}"></span>. </p><p>בנוסף, אפשר להשתמש באינטגרל לחישוב <a href="/wiki/%D7%A0%D7%A4%D7%97" title="נפח">נפח</a> של <a href="/wiki/%D7%92%D7%95%D7%A3_%D7%A1%D7%99%D7%91%D7%95%D7%91" title="גוף סיבוב">גוף סיבוב</a>. גוף סיבוב הוא גוף המתקבל על ידי סיבוב של פונקציה אחת סביב ציר. נפח גוף הסיבוב של הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> סביב ציר x בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \int _{a}^{b}(f(x))^{2}\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \int _{a}^{b}(f(x))^{2}\,dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e731302d957327f87fa88c1ffc23a04dc4e5a9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.335ex; height:6.343ex;" alt="{\displaystyle \pi \int _{a}^{b}(f(x))^{2}\,dx}"></span>. נפח גוף הסיבוב של הפונקציה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> סביב ציר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span>בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \int _{a}^{b}xf(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>x</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \int _{a}^{b}xf(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d809e52cf56a589c0cd1a1fe675622a88a2d1c00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.427ex; height:6.343ex;" alt="{\displaystyle 2\pi \int _{a}^{b}xf(x)\,\mathrm {d} x}"></span>. </p><p><a href="/wiki/%D7%A9%D7%98%D7%97_%D7%A4%D7%A0%D7%99%D7%9D" title="שטח פנים">שטח הפנים</a> של גוף סיבוב הוא (ללא שטח הבסיסים) <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi \int _{a}^{b}f(x){\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi \int _{a}^{b}f(x){\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31077e665e45159275ad75e8d6774f68795b8442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.432ex; height:6.343ex;" alt="{\displaystyle 2\pi \int _{a}^{b}f(x){\sqrt {1+\left(f'(x)\right)^{2}}}\,\mathrm {d} x}"></span>. </p><p>הנפח של הגוף ששטח החתך שלו עבור כל שיעור <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b078651e6d1a522e8955b73059fbd63e13aec616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.882ex; height:2.843ex;" alt="{\displaystyle A(x)}"></span> שווה ל־<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}A(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>A</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}A(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd36900db76eb2c32ea81be61eafa5d6af5c36c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.68ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}A(x)\,\mathrm {d} x}"></span>. </p><p><a href="/wiki/%D7%9E%D7%9E%D7%95%D7%A6%D7%A2#ממוצע_של_פונקציה" title="ממוצע">הערך הממוצע של ערכי הפונקציה</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> בקטע <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"></span> הוא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/134a2a5d3d984dd2ca2124f59ca65341a1282cb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.507ex; height:6.343ex;" alt="{\displaystyle {\frac {1}{b-a}}\int _{a}^{b}f(x)\,\mathrm {d} x}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="נוסחאות_אינטגרציה"><span id=".D7.A0.D7.95.D7.A1.D7.97.D7.90.D7.95.D7.AA_.D7.90.D7.99.D7.A0.D7.98.D7.92.D7.A8.D7.A6.D7.99.D7.94"></span>נוסחאות אינטגרציה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=17" title="עריכת קוד המקור של הפרק: נוסחאות אינטגרציה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=17" title="עריכת פסקה: "נוסחאות אינטגרציה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int 0\,\mathrm {d} x=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int 0\,\mathrm {d} x=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/410bb2f6277fa89e204fd87bfdcf1d6ebf0ddfbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.858ex; height:5.676ex;" alt="{\displaystyle \int 0\,\mathrm {d} x=c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int a\,\mathrm {d} x=ax+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int a\,\mathrm {d} x=ax+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89ef00caf60a24a5c1ebebb971b814e1a115c56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.325ex; height:5.676ex;" alt="{\displaystyle \int a\,\mathrm {d} x=ax+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int x^{n}\,\mathrm {d} x={\frac {x^{n+1}}{n+1}}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int x^{n}\,\mathrm {d} x={\frac {x^{n+1}}{n+1}}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f249657c69cd8eca1c7078c20cb49070d6ab7d9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.318ex; height:6.176ex;" alt="{\displaystyle \int x^{n}\,\mathrm {d} x={\frac {x^{n+1}}{n+1}}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int |x|\mathrm {d} x={\begin{cases}{\frac {x^{2}}{2}}+c&&x\geq 0\\-{\frac {x^{2}}{2}}+c&&x<0\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> </mtd> <mtd /> <mtd> <mi>x</mi> <mo>≥</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>c</mi> </mtd> <mtd /> <mtd> <mi>x</mi> <mo><</mo> <mn>0</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int |x|\mathrm {d} x={\begin{cases}{\frac {x^{2}}{2}}+c&&x\geq 0\\-{\frac {x^{2}}{2}}+c&&x<0\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ac9d2a6a42f3a9afc80cff4ff351962070036f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:32.049ex; height:7.843ex;" alt="{\displaystyle \int |x|\mathrm {d} x={\begin{cases}{\frac {x^{2}}{2}}+c&&x\geq 0\\-{\frac {x^{2}}{2}}+c&&x<0\end{cases}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f(x)g'(x)\,\mathrm {d} x=f(x)g(x)-\int f'(x)g(x)\,\mathrm {d} x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo>∫</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(x)g'(x)\,\mathrm {d} x=f(x)g(x)-\int f'(x)g(x)\,\mathrm {d} x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fc55205bfd0d6cd8eb31ce9af7b0acba214bbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.55ex; height:5.676ex;" alt="{\displaystyle \int f(x)g'(x)\,\mathrm {d} x=f(x)g(x)-\int f'(x)g(x)\,\mathrm {d} x}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {f'(x)g(x)-f(x)g'(x)}{(g(x))^{2}}}\,\mathrm {d} x={\frac {f(x)}{g(x)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {f'(x)g(x)-f(x)g'(x)}{(g(x))^{2}}}\,\mathrm {d} x={\frac {f(x)}{g(x)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60d29fe341cbf10755eebc9a6d454cf676720818" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.377ex; height:6.509ex;" alt="{\displaystyle \int {\frac {f'(x)g(x)-f(x)g'(x)}{(g(x))^{2}}}\,\mathrm {d} x={\frac {f(x)}{g(x)}}}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>±</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>±</mo> <mo>∫</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fa0d5bed88489456c0d7ed8b77245d665b80b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.926ex; height:5.676ex;" alt="{\displaystyle \int (f(x)\pm g(x))dx=\int f(x)dx\pm \int g(x)dx}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int f[g(x)]\cdot g'(x)dx=F[g(x)]+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>⋅</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f[g(x)]\cdot g'(x)dx=F[g(x)]+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e2bc2e77a313b7fae5de1a1e1469b75548f65b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.81ex; height:5.676ex;" alt="{\displaystyle \int f[g(x)]\cdot g'(x)dx=F[g(x)]+c}"></span> כאשר <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> היא פונקציה קדומה של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int e^{x}\,\mathrm {d} x=e^{x}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int e^{x}\,\mathrm {d} x=e^{x}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/717c0946b2fe7d114faac502ae8c1c64a0797ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.048ex; height:5.676ex;" alt="{\displaystyle \int e^{x}\,\mathrm {d} x=e^{x}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \ln x\mathrm {d} x=x\ln x-x+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \ln x\mathrm {d} x=x\ln x-x+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/448887ad8fef04b8e312b854488aa2000b6730fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.348ex; height:5.676ex;" alt="{\displaystyle \int \ln x\mathrm {d} x=x\ln x-x+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln(x)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln(x)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd45cb0d1d25cdf94adb6922b565b1f231b4f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.063ex; height:5.843ex;" alt="{\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln(x)+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {n}{\ln(a)\cdot x}}\,\mathrm {d} x=\log _{a}(x^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {n}{\ln(a)\cdot x}}\,\mathrm {d} x=\log _{a}(x^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e982d55ee4b62da73cde5e6e531cfd5b5057b2a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.943ex; height:6.009ex;" alt="{\displaystyle \int {\frac {n}{\ln(a)\cdot x}}\,\mathrm {d} x=\log _{a}(x^{n})}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {f'(x)}{f(x)}}\,\mathrm {d} x=\ln |f(x)|+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {f'(x)}{f(x)}}\,\mathrm {d} x=\ln |f(x)|+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4dd655d43b15580e777e19d8024e8d075938ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.554ex; height:6.509ex;" alt="{\displaystyle \int {\frac {f'(x)}{f(x)}}\,\mathrm {d} x=\ln |f(x)|+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \cos(x)\,\mathrm {d} x=\sin(x)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \cos(x)\,\mathrm {d} x=\sin(x)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc96d8816122cd689c8783474fe468d6b40f3f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.78ex; height:5.676ex;" alt="{\displaystyle \int \cos(x)\,\mathrm {d} x=\sin(x)+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \sin(x)\,\mathrm {d} x=-\cos(x)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \sin(x)\,\mathrm {d} x=-\cos(x)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bad111d9f92968d229b09852eb42bc45cea921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.976ex; height:5.676ex;" alt="{\displaystyle \int \sin(x)\,\mathrm {d} x=-\cos(x)+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \tan(x)\,\mathrm {d} x=-\ln(|\cos(x)|)+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>tan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \tan(x)\,\mathrm {d} x=-\ln(|\cos(x)|)+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e80517b7d077fec618be51a136259c0c87da57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.909ex; height:5.676ex;" alt="{\displaystyle \int \tan(x)\,\mathrm {d} x=-\ln(|\cos(x)|)+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>arcsin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4d39c58ec59a4b7835ed2559700b2baa7c7e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.006ex; height:5.676ex;" alt="{\displaystyle \int \arcsin(x)\,\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arccos(x)\,\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>arccos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arccos(x)\,\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000bd42231f95a320b7c5f7e84a64c204d7771db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:44.517ex; height:5.676ex;" alt="{\displaystyle \int \arccos(x)\,\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \arctan(x)\,\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln(1+x^{2})+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mi>arctan</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \arctan(x)\,\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln(1+x^{2})+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9caf88367e8def0b7545c9e869e8f5811cf5ec75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:48.824ex; height:5.676ex;" alt="{\displaystyle \int \arctan(x)\,\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln(1+x^{2})+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arcsin {(x)}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>arcsin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arcsin {(x)}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192505d111312eac3bd1ed08949ef97eff9c5851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.571ex; height:6.509ex;" alt="{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arcsin {(x)}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int -{\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arccos {(x)}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>arccos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int -{\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arccos {(x)}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4488228c0bc44021d036d738ce6fc7baa6b13a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:33.634ex; height:6.509ex;" alt="{\displaystyle \int -{\frac {1}{\sqrt {1-x^{2}}}}\,\mathrm {d} x=\arccos {(x)}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\frac {1}{x^{2}+1}}\,\mathrm {d} x=\arctan {(x)}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {1}{x^{2}+1}}\,\mathrm {d} x=\arctan {(x)}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978cee229e1197b9ac31fcd110b8d987375f9701" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.751ex; height:5.676ex;" alt="{\displaystyle \int {\frac {1}{x^{2}+1}}\,\mathrm {d} x=\arctan {(x)}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \sinh {(x)}\,\mathrm {d} x=\cosh {(x)}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \sinh {(x)}\,\mathrm {d} x=\cosh {(x)}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5772d3fb51f27034a80c883cd5d08cb911effc44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.14ex; height:5.676ex;" alt="{\displaystyle \int \sinh {(x)}\,\mathrm {d} x=\cosh {(x)}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \cosh {(x)}\,\mathrm {d} x=\sinh {(x)}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \cosh {(x)}\,\mathrm {d} x=\sinh {(x)}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca9ff96e2c6ac04a96d788f287ae668fb4fb8e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.14ex; height:5.676ex;" alt="{\displaystyle \int \cosh {(x)}\,\mathrm {d} x=\sinh {(x)}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \tanh {(x)}\,\mathrm {d} x=\ln {(\cosh {(x)})}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>tanh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>cosh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \tanh {(x)}\,\mathrm {d} x=\ln {(\cosh {(x)})}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b47113080b6e3b217292cbaa40d26852021fd7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.779ex; height:5.676ex;" alt="{\displaystyle \int \tanh {(x)}\,\mathrm {d} x=\ln {(\cosh {(x)})}+c}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \coth {(x)}\,\mathrm {d} x=\ln {(\sinh {(x)})}+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>coth</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>sinh</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \coth {(x)}\,\mathrm {d} x=\ln {(\sinh {(x)})}+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e3805581e35db35e5c309b8d11eadf94b10ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.264ex; height:5.676ex;" alt="{\displaystyle \int \coth {(x)}\,\mathrm {d} x=\ln {(\sinh {(x)})}+c}"></span> </p><p><br /> </p> <table class="toccolours portalSumm" style="clear:left;float:left; width:250px; margin:0.5em 0.5em 0.5em 0em; border:1px solid #999; padding:5px;"> <tbody><tr> <td> <div style="text-align: center;"> <p><big><b>עיינו גם בפורטל</b></big> </p> <div align="center"> <p><span typeof="mw:File"><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="פורטל:מתמטיקה"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/P_mathematics.svg/90px-P_mathematics.svg.png" decoding="async" width="90" height="81" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/P_mathematics.svg/135px-P_mathematics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/P_mathematics.svg/180px-P_mathematics.svg.png 2x" data-file-width="400" data-file-height="360" /></a></span> </p> </div> <p><b><a href="/wiki/%D7%A4%D7%95%D7%A8%D7%98%D7%9C:%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="פורטל:מתמטיקה">פורטל המתמטיקה</a></b> הוא שער לכל הנושאים הקשורים ב<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה">מתמטיקה</a>. בין היתר, ניתן למצוא בו קישורים אל תחומי המשנה של ענף המתמטיקה, אל מושגי יסוד בתחום, אל ערכים העוסקים בהיסטוריה של המתמטיקה ואל ערכים לגבי מתמטיקאים חשובים. </p> </div> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="ראו_גם"><span id=".D7.A8.D7.90.D7.95_.D7.92.D7.9D"></span>ראו גם</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=18" title="עריכת קוד המקור של הפרק: ראו גם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=18" title="עריכת פסקה: "ראו גם"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" title="שיטות אינטגרציה">שיטות אינטגרציה</a></li> <li><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%91_%D7%9E%D7%9E%D7%93%D7%99" class="mw-redirect" title="אינטגרל רב ממדי">אינטגרל רב ממדי</a></li> <li><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A7%D7%95%D7%95%D7%99" title="אינטגרל קווי">אינטגרל קווי</a></li> <li><a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%91%D7%92" title="אינטגרל לבג">אינטגרל לבג</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=19" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=19" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="sistersitebox plainlinks noprint" style="margin: 0 1em 0.5em 0;float: left;"><tbody><tr><th style="text-align:center">מיזמי <a href="/wiki/%D7%A7%D7%A8%D7%9F_%D7%95%D7%99%D7%A7%D7%99%D7%9E%D7%93%D7%99%D7%94" title="קרן ויקימדיה">קרן ויקימדיה</a></th></tr><tr><td><div class="sisterwikilinkT"><span typeof="mw:File"><span title="ויקימילון"><img alt="ויקימילון" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wiktionary-logo-he.png/20px-Wiktionary-logo-he.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wiktionary-logo-he.png/30px-Wiktionary-logo-he.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Wiktionary-logo-he.png/40px-Wiktionary-logo-he.png 2x" data-file-width="135" data-file-height="135" /></span></span> ערך מילוני בוויקימילון: <b><a href="https://he.wiktionary.org/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" class="extiw" title="wikt:אינטגרל">אינטגרל</a></b></div></td></tr><tr><td><div class="sisterwikilinkT"><span typeof="mw:File"><span title="ויקיספר"><img alt="ויקיספר" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/20px-Wikibooks-logo.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/30px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/40px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> ספר לימוד בוויקיספר: <b><a href="https://he.wikibooks.org/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99/%D7%98%D7%91%D7%9C%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99%D7%9D" class="extiw" title="b:חשבון אינפיניטסימלי/טבלת אינטגרלים">טבלת אינטגרלים</a></b></div></td></tr><tr><td><div class="sisterwikilinkT"><span typeof="mw:File"><span title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> תמונות ומדיה בוויקישיתוף: <b><a href="https://commons.wikimedia.org/wiki/Category:Integration_(mathematics)" class="extiw" title="commons:Category:Integration (mathematics)">אינטגרל</a></b></div></td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r36773993">.mw-parser-output .sistersitebox{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9}</style> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20130325084513/http://integrals.wolfram.com/index.jsp">מחשבון אינטגרלים</a></li> <li><a rel="nofollow" class="external text" href="http://wims.unice.fr/wims/wims.cgi?session=NX0D8EDE31.2&+lang=en&+module=tool%2Fanalysis%2Ffunction.en">מחשבון אינטגרל רימן-סטילטיס</a></li> <li>גדי אלכסנדרוביץ', <a rel="nofollow" class="external text" href="https://gadial.net/2010/11/27/integral/">אז מה זה אינטגרל?</a>, באתר "לא מדויק", 27 בנובמבר 2010</li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Integral.html">אינטגרל</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/integral-mathematics,">topic/integration-mathematics אינטגרל</a>, באתר <a href="/wiki/%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="אנציקלופדיה בריטניקה">אנציקלופדיה בריטניקה</a> <span dir="rtl" class="languageicon">(באנגלית)</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36549940"></li> <li><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/YouTube_full-color_icon_%282017%29.svg/15px-YouTube_full-color_icon_%282017%29.svg.png" decoding="async" width="15" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/YouTube_full-color_icon_%282017%29.svg/23px-YouTube_full-color_icon_%282017%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/09/YouTube_full-color_icon_%282017%29.svg/30px-YouTube_full-color_icon_%282017%29.svg.png 2x" data-file-width="512" data-file-height="358" /></span></span> <a rel="nofollow" class="external text" href="https://youtube.com/watch?v=-xJ-_Emi0Gc">ווייז (waze), שבת ואינטגרל - מה הקשר?</a>, סרטון באתר <a href="/wiki/%D7%99%D7%95%D7%98%D7%99%D7%95%D7%91" title="יוטיוב">יוטיוב</a></li> <li><span typeof="mw:File"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5:National_Library_IL_logo.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/20px-National_Library_IL_logo.svg.png" decoding="async" width="20" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/30px-National_Library_IL_logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/National_Library_IL_logo.svg/40px-National_Library_IL_logo.svg.png 2x" data-file-width="512" data-file-height="397" /></a></span> <a rel="nofollow" class="external text" href="https://www.nli.org.il/he/a-topic/987007555515105171">אינטגרלים</a>, דף שער ב<a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">ספרייה הלאומית</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="הערות_שוליים"><span id=".D7.94.D7.A2.D7.A8.D7.95.D7.AA_.D7.A9.D7.95.D7.9C.D7.99.D7.99.D7.9D"></span>הערות שוליים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&action=edit&section=20" title="עריכת קוד המקור של הפרק: הערות שוליים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C&veaction=edit&section=20" title="עריכת פסקה: "הערות שוליים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist references-small"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">^</a></span> <span class="reference-text"><a href="/wiki/%D7%94%D7%90%D7%A7%D7%93%D7%9E%D7%99%D7%94_%D7%9C%D7%9C%D7%A9%D7%95%D7%9F_%D7%94%D7%A2%D7%91%D7%A8%D7%99%D7%AA" title="האקדמיה ללשון העברית">האקדמיה ללשון העברית</a> קבעה לו את המונח <b>"אַסְכֶּמֶת"</b> (מלשון "סכום"), שלא התקבע.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">^</a></span> <span class="reference-text">אם הגבולות של שתי סדרות נבחרות שונים זה מזה, אז לסדרת הסכומים המתקבלים משילוב הסדרות של חלוקות לסירוגין, פעם זו ופעם זו, לא יהיה גבול.</span> </li> </ol></div></div> <p><br /> </p> <table class="navbox nowraplinks mw-collapsible autocollapse" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA_-_%D7%9E%D7%95%D7%A0%D7%97%D7%99%D7%9D" title="אנליזה מתמטית - מונחים">חשבון אינפיניטסימלי</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="סדרה (מתמטיקה)">סדרה</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%9E%D7%AA%D7%9B%D7%A0%D7%A1%D7%AA" title="סדרה מתכנסת">סדרה מתכנסת</a> • <a href="/wiki/%D7%92%D7%91%D7%95%D7%9C_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="גבול (מתמטיקה)">גבול</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A7%D7%95%D7%A9%D7%99" title="סדרת קושי">סדרת קושי</a> • <a href="/wiki/%D7%98%D7%95%D7%A8_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="טור (מתמטיקה)">טור</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C" title="אינפיניטסימל">אינפיניטסימל</a> • <a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%9E%D7%A9%D7%99%D7%99%D7%9D" title="שדה המספרים הממשיים">שדה המספרים הממשיים</a> • <a href="/wiki/%D7%A2%D7%A8%D7%9A_%D7%9E%D7%95%D7%97%D7%9C%D7%98" title="ערך מוחלט">ערך מוחלט</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%94%D7%9E%D7%A9%D7%95%D7%9C%D7%A9" title="אי-שוויון המשולש">אי-שוויון המשולש</a> • <a href="/wiki/%D7%90%D7%99-%D7%A9%D7%95%D7%95%D7%99%D7%95%D7%9F_%D7%A7%D7%95%D7%A9%D7%99-%D7%A9%D7%95%D7%95%D7%A8%D7%A5" title="אי-שוויון קושי-שוורץ">אי-שוויון קושי-שוורץ</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">פונקציות </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%9E%D7%A9%D7%99%D7%AA" title="פונקציה ממשית">פונקציה</a> • <a href="/wiki/%D7%92%D7%A8%D7%A3_%D7%A9%D7%9C_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94" title="גרף של פונקציה">גרף פונקציה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="פונקציה ליניארית">פונקציה ליניארית</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%95%D7%A0%D7%95%D7%98%D7%95%D7%A0%D7%99%D7%AA" title="פונקציה מונוטונית">פונקציה מונוטונית</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F" title="נקודת קיצון">נקודת קיצון</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%A4%D7%99%D7%AA%D7%95%D7%9C" title="נקודת פיתול">נקודת פיתול</a> •<a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%95%D7%9B%D7%A3" title="נקודת אוכף">נקודת אוכף</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%A2%D7%95%D7%A8%D7%94" title="פונקציה קעורה">פונקציה קעורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A7%D7%9E%D7%95%D7%A8%D7%94" title="פונקציה קמורה">פונקציה קמורה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_(%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94)" title="פונקציה רציפה (אנליזה)">פונקציה רציפה</a> • <a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%A8%D7%A6%D7%99%D7%A4%D7%94_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="פונקציה רציפה במידה שווה">פונקציה רציפה במידה שווה</a> • <a href="/wiki/%D7%A0%D7%A7%D7%95%D7%93%D7%AA_%D7%90%D7%99_%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA" title="נקודת אי רציפות">נקודת אי רציפות</a> • <a href="/wiki/%D7%A0%D7%92%D7%96%D7%A8%D7%AA" title="נגזרת">נגזרת</a> • <a href="/wiki/%D7%98%D7%95%D7%A8_%D7%98%D7%99%D7%99%D7%9C%D7%95%D7%A8" title="טור טיילור">טור טיילור</a> • <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%AA_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA" title="סדרת פונקציות">סדרת פונקציות</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%A0%D7%A7%D7%95%D7%93%D7%AA%D7%99%D7%AA" title="התכנסות נקודתית">התכנסות נקודתית</a> • <a href="/wiki/%D7%94%D7%AA%D7%9B%D7%A0%D7%A1%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="התכנסות במידה שווה">התכנסות במידה שווה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">משפטים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%91%D7%95%D7%9C%D7%A6%D7%90%D7%A0%D7%95-%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפט בולצאנו-ויירשטראס">משפט בולצאנו-ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%95%D7%99%D7%99%D7%A8%D7%A9%D7%98%D7%A8%D7%90%D7%A1" title="משפטי ויירשטראס">משפטי ויירשטראס</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%A0%D7%98%D7%95%D7%A8_%D7%9C%D7%A8%D7%A6%D7%99%D7%A4%D7%95%D7%AA_%D7%91%D7%9E%D7%99%D7%93%D7%94_%D7%A9%D7%95%D7%95%D7%94" title="משפט קנטור לרציפות במידה שווה">משפט קנטור</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A2%D7%A8%D7%9A_%D7%94%D7%91%D7%99%D7%A0%D7%99%D7%99%D7%9D" title="משפט ערך הביניים">משפט ערך הביניים</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A4%D7%A8%D7%9E%D7%94_(%D7%9C%D7%A0%D7%A7%D7%95%D7%93%D7%95%D7%AA_%D7%A7%D7%99%D7%A6%D7%95%D7%9F)" title="משפט פרמה (לנקודות קיצון)">משפט פרמה</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A8%D7%95%D7%9C" title="משפט רול">משפט רול</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27" title="משפט הערך הממוצע של לגראנז'">משפט הערך הממוצע של לגראנז'</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%A2%D7%A8%D7%9A_%D7%94%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%A9%D7%9C_%D7%A7%D7%95%D7%A9%D7%99" title="משפט הערך הממוצע של קושי">משפט הערך הממוצע של קושי</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%93%D7%90%D7%A8%D7%91%D7%95" title="משפט דארבו">משפט דארבו</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A9%D7%A8%D7%A9%D7%A8%D7%AA" title="כלל השרשרת">כלל השרשרת</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%94%D7%A1%D7%A0%D7%93%D7%95%D7%95%D7%99%D7%A5%27" title="כלל הסנדוויץ'">כלל הסנדוויץ'</a> • <a href="/wiki/%D7%9B%D7%9C%D7%9C_%D7%9C%D7%95%D7%A4%D7%99%D7%98%D7%9C" title="כלל לופיטל">כלל לופיטל</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A9%D7%98%D7%95%D7%9C%D7%A5" title="משפט שטולץ">משפט שטולץ</a> • <a href="/wiki/%D7%90%D7%A8%D7%99%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94_%D7%A9%D7%9C_%D7%92%D7%91%D7%95%D7%9C%D7%95%D7%AA" title="אריתמטיקה של גבולות">אריתמטיקה של גבולות</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">האינטגרל </td> <td style="padding-right: 5px; text-align: right;"><a class="mw-selflink selflink">אינטגרל</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%9C%D7%90_%D7%90%D7%9E%D7%99%D7%AA%D7%99" title="אינטגרל לא אמיתי">אינטגרל לא אמיתי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C_%D7%A8%D7%91-%D7%9E%D7%9E%D7%93%D7%99" title="אינטגרל רב-ממדי">אינטגרל רב-ממדי</a> • <a href="/wiki/%D7%94%D7%9E%D7%A9%D7%A4%D7%98_%D7%94%D7%99%D7%A1%D7%95%D7%93%D7%99_%D7%A9%D7%9C_%D7%94%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%94%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99_%D7%95%D7%94%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99" title="המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי">המשפט היסודי של החשבון הדיפרנציאלי והאינטגרלי</a> • <a href="/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94_%D7%91%D7%97%D7%9C%D7%A7%D7%99%D7%9D" title="אינטגרציה בחלקים">אינטגרציה בחלקים</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%95%D7%AA_%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%A6%D7%99%D7%94" title="שיטות אינטגרציה">שיטות אינטגרציה</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">אנליזה מתקדמת </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="פונקציה מרוכבת">פונקציה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A0%D7%99%D7%95%D7%98%D7%95%D7%9F-%D7%A8%D7%A4%D7%A1%D7%95%D7%9F" title="שיטת ניוטון-רפסון">שיטת ניוטון-רפסון</a> • <a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="משוואה דיפרנציאלית">משוואה דיפרנציאלית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> </td></tr> <tr> <td colspan="3" style="background-color: #F2F3F4; text-align: center; font-weight: bold;"><a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%AA" title="אנליזה מתמטית">אנליזה מתמטית</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%95%D7%A7%D7%98%D7%95%D7%A8%D7%99%D7%AA" title="אנליזה וקטורית">אנליזה וקטורית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94" title="טופולוגיה">טופולוגיה</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%AA" title="אנליזה מרוכבת">אנליזה מרוכבת</a> • <a href="/wiki/%D7%90%D7%A0%D7%9C%D7%99%D7%96%D7%94_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%A0%D7%9C%D7%99%D7%AA" title="אנליזה פונקציונלית">אנליזה פונקציונלית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%99%D7%93%D7%94" title="תורת המידה">תורת המידה</a> </td></tr> </tbody></table> <div role="navigation" class="navbox authority-control" aria-labelledby="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q80091?uselang=he&#124;עריכת_הנתון_בוויקינתונים" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q80091?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q80091?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/15px-Blue_pencil_RTL.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/23px-Blue_pencil_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/30px-Blue_pencil_RTL.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-right-width:2px;border-right-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div dir="ltr"> <ul><li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA" title="הספרייה הלאומית">NLI</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007555515105171">987007555515105171</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA_%D7%A9%D7%9C_%D7%A6%D7%A8%D7%A4%D7%AA" title="הספרייה הלאומית של צרפת">BnF</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb119395946">cb119395946</a> <a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb119395946">(data)</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%A9%D7%99%D7%98%D7%AA_%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%AA_%D7%94%D7%A7%D7%95%D7%A0%D7%92%D7%A8%D7%A1" title="שיטת ספריית הקונגרס">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85067099">sh85067099</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA_%D7%A9%D7%9C_%D7%A6%27%D7%9B%D7%99%D7%94" title="הספרייה הלאומית של צ'כיה">NKC</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph121136&CON_LNG=ENG">ph121136</a></span></span></li></ul> </div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">אוחזר מתוך "<a dir="ltr" href="https://he.wikipedia.org/w/index.php?title=אינטגרל&oldid=39127198">https://he.wikipedia.org/w/index.php?title=אינטגרל&oldid=39127198</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94" title="ויקיפדיה:קטגוריה">קטגוריות</a>: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C%D7%99%D7%9D" title="קטגוריה:אינטגרלים">אינטגרלים</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%97%D7%A9%D7%91%D7%95%D7%9F_%D7%90%D7%99%D7%A0%D7%A4%D7%99%D7%A0%D7%99%D7%98%D7%A1%D7%99%D7%9E%D7%9C%D7%99" title="קטגוריה:חשבון אינפיניטסימלי">חשבון אינפיניטסימלי</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">קטגוריות מוסתרות: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="קטגוריה:ערכים עם תבנית בריטניקה">ערכים עם תבנית בריטניקה</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_J9U" title="קטגוריה:ויקיפדיה: ערכים עם מזהה J9U">ויקיפדיה: ערכים עם מזהה J9U</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_BNF" title="קטגוריה:ויקיפדיה: ערכים עם מזהה BNF">ויקיפדיה: ערכים עם מזהה BNF</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_LCCN" title="קטגוריה:ויקיפדיה: ערכים עם מזהה LCCN">ויקיפדיה: ערכים עם מזהה LCCN</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_NKC" title="קטגוריה:ויקיפדיה: ערכים עם מזהה NKC">ויקיפדיה: ערכים עם מזהה NKC</a></li></ul></div></div> </div> </main> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> דף זה נערך לאחרונה ב־29 ביולי 2024, בשעה 08:32.</li> <li id="footer-info-copyright">הטקסט מוגש בכפוף לרישיון <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.he">Creative Commons ייחוס-שיתוף זהה 4.0</a>; ייתכן שישנם תנאים נוספים. ר׳ את <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">תנאי השימוש</a> לפרטים.</li> </ul> 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אינטגרל – ויקיפדיה