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id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Apariencia" > <label id="vector-appearance-dropdown-label" for="vector-appearance-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Apariencia</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" 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title="Menú de usuario" > <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_an.wikipedia.org&uselang=an"><span>Donativos</span></a></li><li id="pt-contribute" class="mw-list-item"><a href="/wiki/Especial:Contribute"><span class="vector-icon mw-ui-icon-edit mw-ui-icon-wikimedia-edit"></span> <span>Contribuir</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Creyar_cuenta&returnto=Integraci%C3%B3n" title="Te recomendamos crear una cuenta e iniciar sesión; sin embargo, no es obligatorio"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Creyar cuenta</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Especial:Encetar_sesi%C3%B3n&returnto=Integraci%C3%B3n" title="Le recomendamos que se rechistre, encara que no ye obligatorio [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Dentrar-ie</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"> Páginas para editores desconectados <a href="/wiki/Aduya:Introducci%C3%B3n" aria-label="Obtenga más información sobre editar"><span>más información</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Especial:A_m%C3%ADa_descusi%C3%B3n" title="Descusión sobre edicions feitas dende ista adreza IP [n]" accesskey="n"><span>Pachina de descusión d'ista IP</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> </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="Sitio"> <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="Contenidos" 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">Contenidos</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mover a la barra lateral</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ocultar</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">Inicio</div> </a> </li> <li id="toc-Integral_indefinita" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integral_indefinita"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Integral indefinita</span> </div> </a> <ul id="toc-Integral_indefinita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Integral_definita" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Integral_definita"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Integral definita</span> </div> </a> <ul id="toc-Integral_definita-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_veiga_tamién" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_veiga_tamién"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Se veiga tamién</span> </div> </a> <ul id="toc-Se_veiga_tamién-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vinclos_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vinclos_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vinclos externos</span> </div> </a> <ul id="toc-Vinclos_externos-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="Contenidos" 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="Cambiar a la tabla de contenidos" > <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">Cambiar a la tabla de contenidos</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">Integración</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="Ir a un artículo en otro idioma. Disponible en 92 idiomas" > <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 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <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 (amárico)" lang="am" hreflang="am" data-title="አጠራቃሚ" data-language-autonym="አማርኛ" data-language-local-name="amárico" class="interlanguage-link-target"><span>አማርኛ</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 (arabe)" lang="ar" hreflang="ar" data-title="تكامل" data-language-autonym="العربية" data-language-local-name="arabe" 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 (asturiano)" lang="ast" hreflang="ast" data-title="Integración" data-language-autonym="Asturianu" data-language-local-name="asturiano" 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 (azerbaiyano)" lang="az" hreflang="az" data-title="İnteqral" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaiyano" 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 (South Azerbaijani)" lang="azb" hreflang="azb" data-title="انتقرال" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" 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 (baskir)" lang="ba" hreflang="ba" data-title="Интеграл" data-language-autonym="Башҡортса" data-language-local-name="baskir" 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 (bielorruso)" lang="be" hreflang="be" data-title="Інтэграл" data-language-autonym="Беларуская" data-language-local-name="bielorruso" 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 (búlgaro)" lang="bg" hreflang="bg" data-title="Интеграл" data-language-autonym="Български" data-language-local-name="búlgaro" 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 (bengalí)" lang="bn" hreflang="bn" data-title="যোগজীকরণ" data-language-autonym="বাংলা" data-language-local-name="bengalí" 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 (bosnio)" lang="bs" hreflang="bs" data-title="Integral" data-language-autonym="Bosanski" data-language-local-name="bosnio" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca badge-Q17437796 badge-featuredarticle mw-list-item" title="articlos destacatos"><a href="https://ca.wikipedia.org/wiki/Integraci%C3%B3" title="Integració (catalán)" lang="ca" hreflang="ca" data-title="Integració" data-language-autonym="Català" data-language-local-name="catalán" 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 (kurdo sorani)" lang="ckb" hreflang="ckb" data-title="تەواوکاری" data-language-autonym="کوردی" data-language-local-name="kurdo sorani" 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 (checo)" lang="cs" hreflang="cs" data-title="Integrál" data-language-autonym="Čeština" data-language-local-name="checo" 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 (chuvasio)" lang="cv" hreflang="cv" data-title="Интеграл" data-language-autonym="Чӑвашла" data-language-local-name="chuvasio" 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 (galés)" lang="cy" hreflang="cy" data-title="Integryn" data-language-autonym="Cymraeg" data-language-local-name="galés" 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 (alemán)" lang="de" hreflang="de" data-title="Integral" data-language-autonym="Deutsch" data-language-local-name="alemán" 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 (Zazaki)" lang="diq" hreflang="diq" data-title="İntegral" data-language-autonym="Zazaki" data-language-local-name="Zazaki" 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 (Central Dusun)" lang="dtp" hreflang="dtp" data-title="Ponompuuvan" data-language-autonym="Kadazandusun" data-language-local-name="Central Dusun" 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 (griego)" lang="el" hreflang="el" data-title="Ολοκλήρωμα" data-language-autonym="Ελληνικά" data-language-local-name="griego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Integral" title="Integral (anglés)" lang="en" hreflang="en" data-title="Integral" data-language-autonym="English" data-language-local-name="anglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Integralo" title="Integralo (esperanto)" lang="eo" hreflang="eo" data-title="Integralo" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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 (espanyol)" lang="es" hreflang="es" data-title="Integración" data-language-autonym="Español" data-language-local-name="espanyol" 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 (estonio)" lang="et" hreflang="et" data-title="Integraal" data-language-autonym="Eesti" data-language-local-name="estonio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu badge-Q17437796 badge-featuredarticle mw-list-item" title="articlos destacatos"><a href="https://eu.wikipedia.org/wiki/Integral" title="Integral (euskera)" lang="eu" hreflang="eu" data-title="Integral" data-language-autonym="Euskara" data-language-local-name="euskera" 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 (persa)" lang="fa" hreflang="fa" data-title="انتگرال" data-language-autonym="فارسی" data-language-local-name="persa" 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 (finés)" lang="fi" hreflang="fi" data-title="Integraali" data-language-autonym="Suomi" data-language-local-name="finés" 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) (francés)" lang="fr" hreflang="fr" data-title="Intégration (mathématiques)" data-language-autonym="Français" data-language-local-name="francés" 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 (gallego)" lang="gl" hreflang="gl" data-title="Integral" data-language-autonym="Galego" data-language-local-name="gallego" 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 (guyaratí)" lang="gu" hreflang="gu" data-title="સંકલન" data-language-autonym="ગુજરાતી" data-language-local-name="guyaratí" 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 (chino hakka)" lang="hak" hreflang="hak" data-title="Chit-fûn-ho̍k" data-language-autonym="客家語 / Hak-kâ-ngî" data-language-local-name="chino hakka" class="interlanguage-link-target"><span>客家語 / Hak-kâ-ngî</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%A0%D7%98%D7%92%D7%A8%D7%9C" title="אינטגרל (hebreo)" lang="he" hreflang="he" data-title="אינטגרל" data-language-autonym="עברית" data-language-local-name="hebreo" class="interlanguage-link-target"><span>עברית</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 (hindi)" lang="hi" hreflang="hi" data-title="समाकलन" data-language-autonym="हिन्दी" data-language-local-name="hindi" 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 (croata)" lang="hr" hreflang="hr" data-title="Integral" data-language-autonym="Hrvatski" data-language-local-name="croata" 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) (criollo haitiano)" lang="ht" hreflang="ht" data-title="Entegrasyon (matematik)" data-language-autonym="Kreyòl ayisyen" data-language-local-name="criollo haitiano" 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 (húngaro)" lang="hu" hreflang="hu" data-title="Integrál" data-language-autonym="Magyar" data-language-local-name="húngaro" 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 (armenio)" lang="hy" hreflang="hy" data-title="Ինտեգրալ" data-language-autonym="Հայերեն" data-language-local-name="armenio" 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 (indonesio)" lang="id" hreflang="id" data-title="Integral" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesio" 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 (ido)" lang="io" hreflang="io" data-title="Integralo" data-language-autonym="Ido" data-language-local-name="ido" 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 (islandés)" lang="is" hreflang="is" data-title="Heildun" data-language-autonym="Íslenska" data-language-local-name="islandés" 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 (italiano)" lang="it" hreflang="it" data-title="Integrale" data-language-autonym="Italiano" data-language-local-name="italiano" 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 (georgiano)" lang="ka" hreflang="ka" data-title="ინტეგრალი" data-language-autonym="ქართული" data-language-local-name="georgiano" 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 (karakalpako)" lang="kaa" hreflang="kaa" data-title="Integral" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpako" 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 (kazajo)" lang="kk" hreflang="kk" data-title="Интеграл" data-language-autonym="Қазақша" data-language-local-name="kazajo" 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 (jemer)" lang="km" hreflang="km" data-title="អាំងតេក្រាល" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="jemer" 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 (canarés)" lang="kn" hreflang="kn" data-title="ಅನುಕಲನಶಾಸ್ತ್ರ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarés" 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 (coreano)" lang="ko" hreflang="ko" data-title="적분" data-language-autonym="한국어" data-language-local-name="coreano" 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 (kurdo)" lang="ku" hreflang="ku" data-title="Întegral" data-language-autonym="Kurdî" data-language-local-name="kurdo" 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 (kirguís)" lang="ky" hreflang="ky" data-title="Интеграл" data-language-autonym="Кыргызча" data-language-local-name="kirguís" 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 (latín)" lang="la" hreflang="la" data-title="Integrale" data-language-autonym="Latina" data-language-local-name="latín" 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 (lombardo)" lang="lmo" hreflang="lmo" data-title="Integral" data-language-autonym="Lombard" data-language-local-name="lombardo" 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 (lituano)" lang="lt" hreflang="lt" data-title="Integralas" data-language-autonym="Lietuvių" data-language-local-name="lituano" 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 (letón)" lang="lv" hreflang="lv" data-title="Integrālis" data-language-autonym="Latviešu" data-language-local-name="letón" 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 (malayálam)" lang="ml" hreflang="ml" data-title="സമാകലനം" data-language-autonym="മലയാളം" data-language-local-name="malayálam" 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 (mongol)" lang="mn" hreflang="mn" data-title="Интеграл" data-language-autonym="Монгол" data-language-local-name="mongol" 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 (maratí)" lang="mr" hreflang="mr" data-title="संकलन" data-language-autonym="मराठी" data-language-local-name="maratí" 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 (malayo)" lang="ms" hreflang="ms" data-title="Kamiran" data-language-autonym="Bahasa Melayu" data-language-local-name="malayo" 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 (maltés)" lang="mt" hreflang="mt" data-title="L-Integral" data-language-autonym="Malti" data-language-local-name="maltés" 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 (birmano)" lang="my" hreflang="my" data-title="အင်တီဂရေးရှင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="birmano" 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 (neerlandés)" lang="nl" hreflang="nl" data-title="Integraal" data-language-autonym="Nederlands" data-language-local-name="neerlandés" 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 (noruego nynorsk)" lang="nn" hreflang="nn" data-title="Integral" data-language-autonym="Norsk nynorsk" data-language-local-name="noruego nynorsk" 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 (noruego bokmal)" lang="nb" hreflang="nb" data-title="Integrasjon" data-language-autonym="Norsk bokmål" data-language-local-name="noruego bokmal" 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 (occitano)" lang="oc" hreflang="oc" data-title="Integracion" data-language-autonym="Occitan" data-language-local-name="occitano" 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 (oromo)" lang="om" hreflang="om" data-title="Wajjummaa" data-language-autonym="Oromoo" data-language-local-name="oromo" 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 (polaco)" lang="pl" hreflang="pl" data-title="Całka" data-language-autonym="Polski" data-language-local-name="polaco" 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 (Western Punjabi)" lang="pnb" hreflang="pnb" data-title="انٹیگرل" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" 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 (portugués)" lang="pt" hreflang="pt" data-title="Integral" data-language-autonym="Português" data-language-local-name="portugués" 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 (rumano)" lang="ro" hreflang="ro" data-title="Integrală" data-language-autonym="Română" data-language-local-name="rumano" 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 (ruso)" lang="ru" hreflang="ru" data-title="Интеграл" data-language-autonym="Русский" data-language-local-name="ruso" 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 (siciliano)" lang="scn" hreflang="scn" data-title="Intiggrali" data-language-autonym="Sicilianu" data-language-local-name="siciliano" 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 (escocés)" lang="sco" hreflang="sco" data-title="Integral" data-language-autonym="Scots" data-language-local-name="escocés" 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 (serbocroata)" lang="sh" hreflang="sh" data-title="Integral" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbocroata" 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 (Simple English)" lang="en-simple" hreflang="en-simple" data-title="Integral" data-language-autonym="Simple English" data-language-local-name="Simple English" 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 (eslovaco)" lang="sk" hreflang="sk" data-title="Integrál" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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 (esloveno)" lang="sl" hreflang="sl" data-title="Integral" data-language-autonym="Slovenščina" data-language-local-name="esloveno" 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 (albanés)" lang="sq" hreflang="sq" data-title="Integrali" data-language-autonym="Shqip" data-language-local-name="albanés" 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 (serbio)" lang="sr" hreflang="sr" data-title="Интеграл" data-language-autonym="Српски / srpski" data-language-local-name="serbio" 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 (sundanés)" lang="su" hreflang="su" data-title="Integral" data-language-autonym="Sunda" data-language-local-name="sundanés" 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 (sueco)" lang="sv" hreflang="sv" data-title="Integral" data-language-autonym="Svenska" data-language-local-name="sueco" 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) (suajili)" lang="sw" hreflang="sw" data-title="Ukamilishaji (hisabati)" data-language-autonym="Kiswahili" data-language-local-name="suajili" 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 (tamil)" lang="ta" hreflang="ta" data-title="தொகையீடு" data-language-autonym="தமிழ்" data-language-local-name="tamil" 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 (tai)" lang="th" hreflang="th" data-title="ปริพันธ์" data-language-autonym="ไทย" data-language-local-name="tai" 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 (turco)" lang="tr" hreflang="tr" data-title="İntegral" data-language-autonym="Türkçe" data-language-local-name="turco" 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 (tártaro)" lang="tt" hreflang="tt" data-title="Интеграл" data-language-autonym="Татарча / tatarça" data-language-local-name="tártaro" 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 (ucraniano)" lang="uk" hreflang="uk" data-title="Інтеграл" data-language-autonym="Українська" data-language-local-name="ucraniano" 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 (urdu)" lang="ur" hreflang="ur" data-title="تکامل" data-language-autonym="اردو" data-language-local-name="urdu" 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 (uzbeko)" lang="uz" hreflang="uz" data-title="Integral" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeko" 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 (Venetian)" lang="vec" hreflang="vec" data-title="Integral" data-language-autonym="Vèneto" data-language-local-name="Venetian" 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 (vietnamita)" lang="vi" hreflang="vi" data-title="Tích phân" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" 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 (valón)" lang="wa" hreflang="wa" data-title="Riveye" data-language-autonym="Walon" data-language-local-name="valón" 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 (chino)" lang="zh" hreflang="zh" data-title="积分" data-language-autonym="中文" data-language-local-name="chino" 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 (chino min nan)" lang="nan" hreflang="nan" data-title="Chek-hun" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="chino min nan" 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 (cantonés)" lang="yue" hreflang="yue" data-title="積分" data-language-autonym="粵語" data-language-local-name="cantonés" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="an" dir="ltr"><table style="margin: 0 auto; width:80%; background-color:var(--background-color-neutral-subtle, #f8f9fa); color:inherit; border:1px solid var(--border-color-muted, #dadde3); padding:4px; text-align: center; font-size:85%;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/Imachen:Translate_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Translate_icon.svg/35px-Translate_icon.svg.png" decoding="async" width="35" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Translate_icon.svg/53px-Translate_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/Translate_icon.svg/70px-Translate_icon.svg.png 2x" data-file-width="243" data-file-height="235" /></a></span> </td> <td>Iste articlo ye en proceso de 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Puez aduyar a completar este proceso revisando l'articlo, fendo-ie los cambios ortograficos necesarios y sacando dimpués ista plantilla. </td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Imachen:RightRiemann2.PNG" class="mw-file-description" title="Integral de Riemann"><img alt="Integral de Riemann" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/250px-RightRiemann2.PNG" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/375px-RightRiemann2.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RightRiemann2.PNG/500px-RightRiemann2.PNG 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Integral de Riemann</figcaption></figure> <p>En <a href="/wiki/Matematicas" title="Matematicas">matematicas</a>, a <b>integración</b> se relaciona con dos problemas clasicos de l'<a href="/wiki/Anal%C3%ADs_matematica" title="Analís matematica">analís matematica</a>, que no pareixen relacionatos: </p> <ul><li>O <a href="/wiki/Calculo" title="Calculo">calculo</a> d'arias y volumens, y l'acción que una <a href="/wiki/Funci%C3%B3n_matematica" title="Función matematica">función</a> d'una u cuantas variables les aplica a las rechions debant ditas.</li> <li>A obtención d'a <b>primitiva</b> d'una <a href="/wiki/Funci%C3%B3n_matematica" title="Función matematica">función</a>, isto ye, trobar a función que a suya derivata ye a función dada, estando a "operación inversa" a la <a href="/w/index.php?title=Derivaci%C3%B3n&action=edit&redlink=1" class="new" title="Derivación (a pachina encara no existe)">derivación</a></li></ul> <p>Os estudios de <a href="/w/index.php?title=Isaac_Barrow&action=edit&redlink=1" class="new" title="Isaac Barrow (a pachina encara no existe)">Isaac Barrow</a>, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a> y <a href="/wiki/Gottfried_Leibniz" title="Gottfried Leibniz">Gottfried Leibniz</a>, dioron forma a lo <a href="/w/index.php?title=Teorema_fundamental_d%27o_calculo&action=edit&redlink=1" class="new" title="Teorema fundamental d'o calculo (a pachina encara no existe)">teorema fundamental d'o calculo</a>, que amuestra la intima relación en a solución d'istos dos problemas. Se clama <a href="/w/index.php?title=Integraci%C3%B3n_definita&action=edit&redlink=1" class="new" title="Integración definita (a pachina encara no existe)">integración definita</a> a la obtención de l'aria baixo una curva, y <a href="/w/index.php?title=Integraci%C3%B3n_indefinita&action=edit&redlink=1" class="new" title="Integración indefinita (a pachina encara no existe)">integración indefinita</a> a la operación inversa d'a derivación. Tamién se diz <b>integración</b> a la resolución d'una <a href="/wiki/Ecuaci%C3%B3n_diferencial" title="Ecuación diferencial">ecuación diferencial</a>, una ecuación en a que a incognita ye una u cuantas funcions y as suyas derivatas. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Integral_indefinita">Integral indefinita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integraci%C3%B3n&veaction=edit&section=1" title="Editar a sección: Integral indefinita" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Integraci%C3%B3n&action=edit&section=1" title="Editar el código fuente de la sección: Integral indefinita"><span>modificar o codigo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dada una <a href="/wiki/Funci%C3%B3n_matematica" title="Función matematica">función</a> <b>F(x)</b> tal que a suya derivata ye <b>F'(x) = f(x)</b>, allora decimos que <b>F</b> ye a <b>integral</b> u <b>primitiva</b> de <b>f</b>, definindo asinas a integración como a inversa d'a derivación. Simbolicament se denota por </p> <center> <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 f(x)dx}"> <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> <mo>∫</mo> <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(x)=\int f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfb9b69f32d23eb748c4f6c921cb81a4e1ef23e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.522ex; height:5.676ex;" alt="{\displaystyle F(x)=\int f(x)dx}"></span> </center> <p>Una función dada no tien una sola integral indefinita. Por eixemplo, ta la función <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)=x+2}"> <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>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa206193789f7ab8eb20f74f154310b2f199b587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.849ex; height:2.843ex;" alt="{\displaystyle f(x)=x+2}"></span>, as siguients funcions son todas primitivas d'a mesma: </p> <center><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 {\begin{matrix}F_{1}(x)={\frac {1}{2}}x^{2}+2x\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+{\frac {3}{2}}\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+\pi \end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>π</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}F_{1}(x)={\frac {1}{2}}x^{2}+2x\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+{\frac {3}{2}}\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+\pi \end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e9606bd80cb0f81e24fbade50a41192a6d72a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.012ex; margin-bottom: -0.326ex; width:23.411ex; height:11.843ex;" alt="{\displaystyle {\begin{matrix}F_{1}(x)={\frac {1}{2}}x^{2}+2x\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+{\frac {3}{2}}\\F_{1}(x)={\frac {1}{2}}x^{2}+2x+\pi \end{matrix}}}"></span></center> <p>En cheneral, si <b>F(x)</b> ye una primitiva de <b>f(x)</b>, allora cualsiquier función d'a forma <b>F(x)+c</b>, estando <b>c</b> una constant cualsiquiera, ye tamién una primitiva de <b>f(x)</b>. </p> <div class="mw-heading mw-heading2"><h2 id="Integral_definita">Integral definita</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Integraci%C3%B3n&veaction=edit&section=2" title="Editar a sección: Integral definita" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Integraci%C3%B3n&action=edit&section=2" title="Editar el código fuente de la sección: Integral definita"><span>modificar o codigo</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File"><a href="/wiki/Imachen:Integral_as_region_under_curve.png" class="mw-file-description" title="Integral definita, vista como l'aria d'a rechión baixo una curva"><img alt="Integral definita, vista como l'aria d'a rechión baixo una curva" 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>Integral definita, vista como l'aria d'a rechión baixo una curva</figcaption></figure> <p>Dada una función: </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 y=f(x)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=f(x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceab146b2e6d4d4a3580db2ba2d1d240edcbc080" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.059ex; height:2.843ex;" alt="{\displaystyle y=f(x)\,}"></span></dd></dl> <p>se define a <b>integral definita entre a y b d'a función f</b> como l'aria <b>S</b> y se denota por: </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 S=\int _{a}^{b}f(x)dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <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> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=\int _{a}^{b}f(x)dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701e867eafddbd27baec4b24cd02f8045c34dba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.35ex; height:6.343ex;" alt="{\displaystyle S=\int _{a}^{b}f(x)dx}"></span></dd></dl> <p>Si se tien una primitiva (integral indefinita) d'a función f: </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 f(x)dx=F(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> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int f(x)dx=F(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7ebea5842218823563b6f0a85a6c3a4f79d99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.522ex; height:5.676ex;" alt="{\displaystyle \int f(x)dx=F(x)}"></span></dd></dl> <p>allora </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 S=F(b)-F(a)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=F(b)-F(a)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3aada020fd60d007a082620c80758d39618abb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.153ex; height:2.843ex;" alt="{\displaystyle S=F(b)-F(a)\,}"></span></dd></dl> <p>a ista relación entre a integral indefinita y a superficie baixo a función le se clama <a href="/w/index.php?title=Teorema_fundamental_d%27o_calculo_integral&action=edit&redlink=1" class="new" title="Teorema fundamental d'o calculo integral (a pachina encara no existe)">Teorema fundamental d'o calculo integral</a>. </p><p>O desarrollo d'a <a href="/w/index.php?title=Teor%C3%ADa_d%27a_mida&action=edit&redlink=1" class="new" title="Teoría d'a mida (a pachina encara no existe)">Teoría d'a mida</a> en a Matematica ha produciu historicament diversas definicions d'o concepto d'integral </p> <ul><li>A <a 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