Sustitución de Euler - Wikipedia, la enciclopedia libre

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href="#Primera_sustitución"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Primera sustitución</span> </div> </a> <ul id="toc-Primera_sustitución-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Segunda_sustitución" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Segunda_sustitución"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Segunda sustitución</span> </div> </a> <ul id="toc-Segunda_sustitución-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tercera_sustitución" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tercera_sustitución"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Tercera sustitución</span> </div> </a> <ul id="toc-Tercera_sustitución-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ejemplos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ejemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Ejemplos</span> </div> </a> <button aria-controls="toc-Ejemplos-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>Alternar subsección Ejemplos</span> </button> <ul id="toc-Ejemplos-sublist" class="vector-toc-list"> <li id="toc-Primera_sustitución_de_Euler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primera_sustitución_de_Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Primera sustitución de Euler</span> </div> </a> <ul id="toc-Primera_sustitución_de_Euler-sublist" class="vector-toc-list"> <li id="toc-Ejemplo_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ejemplo_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>Ejemplo 1</span> </div> </a> <ul id="toc-Ejemplo_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ejemplo_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ejemplo_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>Ejemplo 2</span> </div> </a> <ul id="toc-Ejemplo_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Segunda_sustitución_de_Euler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Segunda_sustitución_de_Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Segunda sustitución de Euler</span> </div> </a> <ul id="toc-Segunda_sustitución_de_Euler-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tercera_sustitución_de_Euler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tercera_sustitución_de_Euler"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Tercera sustitución de Euler</span> </div> </a> <ul id="toc-Tercera_sustitución_de_Euler-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizaciones" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizaciones"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Generalizaciones</span> </div> </a> <ul id="toc-Generalizaciones-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Véase_también" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Véase_también"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Véase también</span> </div> </a> <ul id="toc-Véase_también-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referencias" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referencias"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referencias</span> </div> </a> <ul id="toc-Referencias-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Enlaces_externos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Enlaces_externos"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Enlaces externos</span> </div> </a> <ul id="toc-Enlaces_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">Sustitución de Euler</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 7 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-7" 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">7 idiomas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Eulersche_Substitution" title="Eulersche Substitution (alemán)" lang="de" hreflang="de" data-title="Eulersche Substitution" data-language-autonym="Deutsch" data-language-local-name="alemán" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Euler_substitution" title="Euler substitution (inglés)" lang="en" hreflang="en" data-title="Euler substitution" data-language-autonym="English" data-language-local-name="inglés" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%BA%DB%8C%DB%8C%D8%B1_%D9%85%D8%AA%D8%BA%DB%8C%D8%B1_%D8%A7%D9%88%DB%8C%D9%84%D8%B1" title="تغییر متغیر اویلر (persa)" lang="fa" hreflang="fa" data-title="تغییر متغیر اویلر" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sostituzioni_di_Eulero" title="Sostituzioni di Eulero (italiano)" lang="it" hreflang="it" data-title="Sostituzioni di Eulero" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%98%A4%EC%9D%BC%EB%9F%AC_%EC%B9%98%ED%99%98" title="오일러 치환 (coreano)" lang="ko" hreflang="ko" data-title="오일러 치환" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D1%81%D1%82%D0%B0%D0%BD%D0%BE%D0%B2%D0%BA%D0%B8_%D0%AD%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Подстановки Эйлера (ruso)" lang="ru" hreflang="ru" data-title="Подстановки Эйлера" data-language-autonym="Русский" data-language-local-name="ruso" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D1%81%D1%82%D0%B0%D0%BD%D0%BE%D0%B2%D0%BA%D0%B0_%D0%95%D0%B9%D0%BB%D0%B5%D1%80%D0%B0" title="Підстановка Ейлера (ucraniano)" lang="uk" hreflang="uk" data-title="Підстановка Ейлера" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</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/Q4368359#sitelinks-wikipedia" title="Editar enlaces interlingüísticos" class="wbc-editpage">Editar enlaces</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espacios de nombres"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Herramientas</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.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-page-tools.unpin">ocultar</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Más opciones" > <div 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todas las páginas de la wiki que enlazan aquí [j]" accesskey="j"><span>Lo que enlaza aquí</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Especial:CambiosEnEnlazadas/Sustituci%C3%B3n_de_Euler" rel="nofollow" title="Cambios recientes en las páginas que enlazan con esta [k]" accesskey="k"><span>Cambios en enlazadas</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=es" title="Subir archivos [u]" accesskey="u"><span>Subir archivo</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Especial:P%C3%A1ginasEspeciales" title="Lista de todas las páginas especiales [q]" accesskey="q"><span>Páginas especiales</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&oldid=134575777" title="Enlace permanente a esta versión de la página"><span>Enlace permanente</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=info" title="Más información sobre esta página"><span>Información de la página</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Especial:Citar&page=Sustituci%C3%B3n_de_Euler&id=134575777&wpFormIdentifier=titleform" title="Información sobre cómo citar esta página"><span>Citar esta página</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Especial:Acortador_de_URL&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FSustituci%25C3%25B3n_de_Euler"><span>Obtener URL acortado</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Especial:QrCode&url=https%3A%2F%2Fes.wikipedia.org%2Fwiki%2FSustituci%25C3%25B3n_de_Euler"><span>Descargar código QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Imprimir/exportar </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Especial:Libro&bookcmd=book_creator&referer=Sustituci%C3%B3n+de+Euler"><span>Crear un libro</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Especial:DownloadAsPdf&page=Sustituci%C3%B3n_de_Euler&action=show-download-screen"><span>Descargar como PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&printable=yes" title="Versión imprimible de esta página [p]" accesskey="p"><span>Versión para imprimir</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> En otros proyectos </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q4368359" title="Enlace al elemento conectado del repositorio de datos [g]" accesskey="g"><span>Elemento de Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Página de herramientas"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Apariencia"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Apariencia</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.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-appearance.unpin">ocultar</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">De Wikipedia, la enciclopedia libre</div> </div> <div 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="es" dir="ltr"><p>La <b>sustitución de Euler</b> es un método para evaluar integrales de la forma </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 R(x,{\sqrt {ax^{2}+bx+c}})\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de8039552da1359e55122a07d4a3875b8f33c88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.05ex; height:5.676ex;" alt="{\displaystyle \int R(x,{\sqrt {ax^{2}+bx+c}})\,dx,}"></span></dd></dl> <p>donde <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> es una función racional de <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> y de <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 {\sqrt {ax^{2}+bx+c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11567a8ebd2e19291b9c733b1781fed116483262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.952ex; height:3.509ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}}"></span>. En tales casos, el integrando se puede cambiar a una función racional usando las sustituciones de Euler.<sup id="cite_ref-1" class="reference separada"><a href="#cite_note-1"><span class="corchete-llamada">[</span>1<span class="corchete-llamada">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Primera_sustitución"><span id="Primera_sustituci.C3.B3n"></span>Primera sustitución</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=1" title="Editar sección: Primera sustitución"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>La primera sustitución de Euler se utiliza cuando <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span>. Se sustituye </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 {\sqrt {ax^{2}+bx+c}}=\pm x{\sqrt {a}}+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mo>±</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}=\pm x{\sqrt {a}}+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d96c437d96aa9e353c74184cac4c63b35ac87bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.034ex; height:3.676ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}=\pm x{\sqrt {a}}+t}"></span></dd></dl> <p>y se resuelve la expresión resultante para <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>. Se tiene que </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 x={\frac {c-t^{2}}{\pm 2t{\sqrt {a}}-b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo>±</mo> <mn>2</mn> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>−</mo> <mi>b</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {c-t^{2}}{\pm 2t{\sqrt {a}}-b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff71e6447253ad4ebc8133bbe176d4c7de613da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:16.078ex; height:6.676ex;" alt="{\displaystyle x={\frac {c-t^{2}}{\pm 2t{\sqrt {a}}-b}}}"></span></dd></dl> <p>y el término <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 dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845c817e348381a13f3fad5184169ce0e021c685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}"></span> se puede expresar racionalmente en <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>. </p><p>En esta sustitución, se puede elegir el signo positivo o el signo negativo. </p> <div class="mw-heading mw-heading2"><h2 id="Segunda_sustitución"><span id="Segunda_sustituci.C3.B3n"></span>Segunda sustitución</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=2" title="Editar sección: Segunda sustitución"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si <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>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ba126f626d61752f62eaacaf11761a54de4dc84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c>0}"></span>, se toma </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 {\sqrt {ax^{2}+bx+c}}=xt\pm {\sqrt {c}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mi>x</mi> <mi>t</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}=xt\pm {\sqrt {c}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a171a5ce7034ed4150b72f411d7ad6d1504bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.65ex; height:3.676ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}=xt\pm {\sqrt {c}}.}"></span></dd></dl> <p>Se resuelve para <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> de manera similar al caso anterior y entonces </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 x={\frac {\pm 2t{\sqrt {c}}-b}{a-t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>±</mo> <mn>2</mn> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>c</mi> </msqrt> </mrow> <mo>−</mo> <mi>b</mi> </mrow> <mrow> <mi>a</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {\pm 2t{\sqrt {c}}-b}{a-t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130291e8fb094109982f1f15fb568da419102d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.502ex; height:6.343ex;" alt="{\displaystyle x={\frac {\pm 2t{\sqrt {c}}-b}{a-t^{2}}}.}"></span></dd></dl> <p>Nuevamente, se puede elegir el signo positivo o negativo. </p> <div class="mw-heading mw-heading2"><h2 id="Tercera_sustitución"><span id="Tercera_sustituci.C3.B3n"></span>Tercera sustitución</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=3" title="Editar sección: Tercera sustitución"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Si el polinomio <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 ax^{2}+bx+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126c6935d3dd9f1c1da0c388ca2799be4f6f237c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.629ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c}"></span> tiene raíces reales <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> y <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>, se puede elegir </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 {\sqrt {ax^{2}+bx+c}}={\sqrt {a(x-\alpha )(x-\beta )}}=(x-\alpha )t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}={\sqrt {a(x-\alpha )(x-\beta )}}=(x-\alpha )t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a984939885869080cbdf61d5d110f37c17a96eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.788ex; height:4.843ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}={\sqrt {a(x-\alpha )(x-\beta )}}=(x-\alpha )t}"></span>.</dd></dl> <p>Esto produce </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 x={\frac {a\beta -\alpha t^{2}}{a-t^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mi>β</mi> <mo>−</mo> <mi>α</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>a</mi> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {a\beta -\alpha t^{2}}{a-t^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b496cbd58b65d8422f0b432df8dc9cf39dce4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.695ex; height:6.176ex;" alt="{\displaystyle x={\frac {a\beta -\alpha t^{2}}{a-t^{2}}},}"></span></dd></dl> <p>y como en los casos anteriores, se puede expresar el integrando entero racionalmente en <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>. </p> <div class="mw-heading mw-heading2"><h2 id="Ejemplos">Ejemplos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=4" title="Editar sección: Ejemplos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Primera_sustitución_de_Euler"><span id="Primera_sustituci.C3.B3n_de_Euler"></span>Primera sustitución de Euler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=5" title="Editar sección: Primera sustitución de Euler"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Ejemplo_1">Ejemplo 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=6" title="Editar sección: Ejemplo 1"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En la integral </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 \!{\frac {\ dx}{\sqrt {x^{2}+c}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \!{\frac {\ dx}{\sqrt {x^{2}+c}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f92349a5b13d1453cd11211d16da4efa2c440a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.585ex; height:6.676ex;" alt="{\displaystyle \int \!{\frac {\ dx}{\sqrt {x^{2}+c}}}}"></span></dd></dl> <p>se puede usar la primera sustitución y establecer <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 {\sqrt {x^{2}+c}}=-x+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+c}}=-x+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d777a194bb49c80469a89353c7f7703b55499c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.471ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}+c}}=-x+t}"></span>, así </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 x={\frac {t^{2}-c}{2t}}\quad \quad \ dx={\frac {t^{2}+c}{2t^{2}}}\,\ dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="1em" /> <mspace width="1em" /> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {t^{2}-c}{2t}}\quad \quad \ dx={\frac {t^{2}+c}{2t^{2}}}\,\ dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9dc9f517e8dfeb612a450bd80cd0644cb9d82a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.476ex; height:6.009ex;" alt="{\displaystyle x={\frac {t^{2}-c}{2t}}\quad \quad \ dx={\frac {t^{2}+c}{2t^{2}}}\,\ dt}"></span></dd> <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 {\sqrt {x^{2}+c}}=-{\frac {t^{2}-c}{2t}}+t={\frac {t^{2}+c}{2t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+c}}=-{\frac {t^{2}-c}{2t}}+t={\frac {t^{2}+c}{2t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed1f32a86d6447dcbf71d0b3bfef6aeb718c20b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.394ex; height:5.676ex;" alt="{\displaystyle {\sqrt {x^{2}+c}}=-{\frac {t^{2}-c}{2t}}+t={\frac {t^{2}+c}{2t}}}"></span></dd></dl> <p>En consecuencia, se obtiene: </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 {\frac {\ dx}{\sqrt {x^{2}+c}}}=\int {\frac {\frac {t^{2}+c}{2t^{2}}}{\frac {t^{2}+c}{2t}}}\,\ dt=\int \!{\frac {\ dt}{t}}=\ln |t|+C=\ln |x+{\sqrt {x^{2}+c}}|+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </mrow> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> </mrow> <mi>t</mi> </mfrac> </mrow> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>C</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <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 {\ dx}{\sqrt {x^{2}+c}}}=\int {\frac {\frac {t^{2}+c}{2t^{2}}}{\frac {t^{2}+c}{2t}}}\,\ dt=\int \!{\frac {\ dt}{t}}=\ln |t|+C=\ln |x+{\sqrt {x^{2}+c}}|+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e7535c350cfd2aab7a9aa053895f717e1d9313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:70.742ex; height:9.176ex;" alt="{\displaystyle \int {\frac {\ dx}{\sqrt {x^{2}+c}}}=\int {\frac {\frac {t^{2}+c}{2t^{2}}}{\frac {t^{2}+c}{2t}}}\,\ dt=\int \!{\frac {\ dt}{t}}=\ln |t|+C=\ln |x+{\sqrt {x^{2}+c}}|+C}"></span></dd></dl> <p>Con <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=\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf5f2e1fb1c0a54282f2e1ee02301cccc95e82e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.076ex; height:2.176ex;" alt="{\displaystyle c=\pm 1}"></span> se obtienen las fórmulas </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 {\begin{aligned}\int {\frac {\ dx}{\sqrt {x^{2}+1}}}&={\mbox{arsinh}}(x)+C\\[6pt]\int {\frac {\ dx}{\sqrt {x^{2}-1}}}&={\mbox{arcosh}}(x)+C\qquad (x>1)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>arsinh</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>arcosh</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int {\frac {\ dx}{\sqrt {x^{2}+1}}}&={\mbox{arsinh}}(x)+C\\[6pt]\int {\frac {\ dx}{\sqrt {x^{2}-1}}}&={\mbox{arcosh}}(x)+C\qquad (x>1)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba7bf29d04c278d8e2783f7a6e67c3b8a143776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:42.246ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\int {\frac {\ dx}{\sqrt {x^{2}+1}}}&={\mbox{arsinh}}(x)+C\\[6pt]\int {\frac {\ dx}{\sqrt {x^{2}-1}}}&={\mbox{arcosh}}(x)+C\qquad (x>1)\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Ejemplo_2">Ejemplo 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=7" title="Editar sección: Ejemplo 2"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para encontrar el valor de </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 {\frac {1}{x{\sqrt {x^{2}+4x-4}}}}dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {1}{x{\sqrt {x^{2}+4x-4}}}}dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7e8d8283800eb90363da59ccc5267320f03a05a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.982ex; height:6.509ex;" alt="{\displaystyle \int {\frac {1}{x{\sqrt {x^{2}+4x-4}}}}dx,}"></span></dd></dl> <p>se determina <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> usando la primera sustitución de Euler, <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 {\sqrt {x^{2}+4x-4}}={\sqrt {1}}x+t=x+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> </msqrt> </mrow> <mi>x</mi> <mo>+</mo> <mi>t</mi> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {x^{2}+4x-4}}={\sqrt {1}}x+t=x+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92e92b84a6e6a9c37e9d2453634c82124369027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.358ex; height:3.509ex;" alt="{\displaystyle {\sqrt {x^{2}+4x-4}}={\sqrt {1}}x+t=x+t}"></span>. Al elevar al cuadrado ambos lados de la ecuación se obtiene <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}+4x-4=x^{2}+2xt+t^{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> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>t</mi> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+4x-4=x^{2}+2xt+t^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/966c454107c026844f2ff3bdcba55625925af351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.108ex; height:2.843ex;" alt="{\displaystyle x^{2}+4x-4=x^{2}+2xt+t^{2}}"></span>, a partir de lo que los términos en <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> se cancelan. Resolviendo la ecuación, se obtiene <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> </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 x={\frac {t^{2}+4}{4-2t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mrow> <mrow> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {t^{2}+4}{4-2t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/084c2ae6034b546bbd505ddb27c23a796b23f122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.916ex; height:5.843ex;" alt="{\displaystyle x={\frac {t^{2}+4}{4-2t}}.}"></span></dd></dl> <p>A partir de ahí, resulta que los diferenciales <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 dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/845c817e348381a13f3fad5184169ce0e021c685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}"></span> y <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 dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee76a835701fd1f26047a09855f2ea36bb08fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.055ex; height:2.176ex;" alt="{\displaystyle dt}"></span> están relacionados 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 dx={\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>t</mi> <mo>+</mo> <mn>8</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx={\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56ec223693085f3895a77ea3e31f43f6f3f121ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.892ex; height:6.509ex;" alt="{\displaystyle dx={\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}dt.}"></span></dd></dl> <p>Por lo tanto, </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 {\begin{aligned}\int {\frac {dx}{x{\sqrt {x^{2}+4x-4}}}}&=\int {\frac {\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}{({\frac {t^{2}+4}{4-2t}})({\frac {-t^{2}+4t+4}{4-2t}})}}dt\\[6pt]&=2\int {\frac {dt}{t^{2}+4}}=\tan ^{-1}\left({\frac {t}{2}}\right)+C&&t={\sqrt {x^{2}+4x-4}}-x\\[6pt]&=\tan ^{-1}\left({\frac {{\sqrt {x^{2}+4x-4}}-x}{2}}\right)+C\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>t</mi> <mo>+</mo> <mn>8</mn> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mrow> <mrow> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>t</mi> <mo>+</mo> <mn>4</mn> </mrow> <mrow> <mn>4</mn> <mo>−</mo> <mn>2</mn> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>t</mi> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> <mtd /> <mtd> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </msqrt> </mrow> <mo>−</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>tan</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </msqrt> </mrow> <mo>−</mo> <mi>x</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int {\frac {dx}{x{\sqrt {x^{2}+4x-4}}}}&=\int {\frac {\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}{({\frac {t^{2}+4}{4-2t}})({\frac {-t^{2}+4t+4}{4-2t}})}}dt\\[6pt]&=2\int {\frac {dt}{t^{2}+4}}=\tan ^{-1}\left({\frac {t}{2}}\right)+C&&t={\sqrt {x^{2}+4x-4}}-x\\[6pt]&=\tan ^{-1}\left({\frac {{\sqrt {x^{2}+4x-4}}-x}{2}}\right)+C\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/416135ec0f98e3c50b1a8c6dc10d4073211c791b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.565ex; margin-bottom: -0.273ex; width:82.466ex; height:26.843ex;" alt="{\displaystyle {\begin{aligned}\int {\frac {dx}{x{\sqrt {x^{2}+4x-4}}}}&=\int {\frac {\frac {-2t^{2}+8t+8}{(4-2t)^{2}}}{({\frac {t^{2}+4}{4-2t}})({\frac {-t^{2}+4t+4}{4-2t}})}}dt\\[6pt]&=2\int {\frac {dt}{t^{2}+4}}=\tan ^{-1}\left({\frac {t}{2}}\right)+C&&t={\sqrt {x^{2}+4x-4}}-x\\[6pt]&=\tan ^{-1}\left({\frac {{\sqrt {x^{2}+4x-4}}-x}{2}}\right)+C\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Segunda_sustitución_de_Euler"><span id="Segunda_sustituci.C3.B3n_de_Euler"></span>Segunda sustitución de Euler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=8" title="Editar sección: Segunda sustitución de Euler"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En la integral </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 \!{\frac {dx}{x{\sqrt {-x^{2}+x+2}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \!{\frac {dx}{x{\sqrt {-x^{2}+x+2}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54702557947f3719d0a492169c4a26740acbbac0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.695ex; height:6.676ex;" alt="{\displaystyle \int \!{\frac {dx}{x{\sqrt {-x^{2}+x+2}}}},}"></span></dd></dl> <p>se puede usar la segunda sustitución y configurar <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 {\sqrt {-x^{2}+x+2}}=xt+{\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mi>x</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x^{2}+x+2}}=xt+{\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d01064f9ddce7d946cb0e9a5c7cbff3bf780d54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.895ex; height:3.509ex;" alt="{\displaystyle {\sqrt {-x^{2}+x+2}}=xt+{\sqrt {2}}}"></span>. Así </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 x={\frac {1-2{\sqrt {2}}t}{t^{2}+1}}\qquad dx={\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>t</mi> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {1-2{\sqrt {2}}t}{t^{2}+1}}\qquad dx={\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777405e6af1c688d2bb755e94e7203b18c9b1aed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:46.293ex; height:6.676ex;" alt="{\displaystyle x={\frac {1-2{\sqrt {2}}t}{t^{2}+1}}\qquad dx={\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}dt,}"></span></dd></dl> <p>y </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 {\sqrt {-x^{2}+x+2}}={\frac {1-2{\sqrt {2t}}}{t^{2}+1}}t+{\sqrt {2}}={\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>t</mi> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x^{2}+x+2}}={\frac {1-2{\sqrt {2t}}}{t^{2}+1}}t+{\sqrt {2}}={\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd2fd25f58b82240e4ac24367d92d362c5c6913" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.859ex; height:6.343ex;" alt="{\displaystyle {\sqrt {-x^{2}+x+2}}={\frac {1-2{\sqrt {2t}}}{t^{2}+1}}t+{\sqrt {2}}={\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}"></span></dd></dl> <p>En consecuencia, se obtiene: </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 {\begin{aligned}\int {\frac {dx}{x{\sqrt {-x^{2}+x+2}}}}&=\int {\frac {\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}{{\frac {1-2{\sqrt {2}}t}{t^{2}+1}}{\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}}\;dt\\&=\int \!{\frac {-2}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\int {\frac {-2{\sqrt {2}}}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\ln {\Biggl |}2{\sqrt {2}}t-1{\Biggl |}+C\\&={\frac {\sqrt {2}}{2}}\ln {\Biggl |}2{\sqrt {2}}{\frac {{\sqrt {-x^{2}+x+2}}-{\sqrt {2}}}{x}}-1{\Biggl |}+C\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mfrac> <mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>t</mi> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mi>x</mi> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.470em" minsize="2.470em">|</mo> </mrow> </mrow> <mo>+</mo> <mi>C</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int {\frac {dx}{x{\sqrt {-x^{2}+x+2}}}}&=\int {\frac {\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}{{\frac {1-2{\sqrt {2}}t}{t^{2}+1}}{\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}}\;dt\\&=\int \!{\frac {-2}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\int {\frac {-2{\sqrt {2}}}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\ln {\Biggl |}2{\sqrt {2}}t-1{\Biggl |}+C\\&={\frac {\sqrt {2}}{2}}\ln {\Biggl |}2{\sqrt {2}}{\frac {{\sqrt {-x^{2}+x+2}}-{\sqrt {2}}}{x}}-1{\Biggl |}+C\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a2119d036458fa5d0db5382fd7b049ead1f0602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -18.338ex; width:65.561ex; height:37.843ex;" alt="{\displaystyle {\begin{aligned}\int {\frac {dx}{x{\sqrt {-x^{2}+x+2}}}}&=\int {\frac {\frac {2{\sqrt {2}}t^{2}-2t-2{\sqrt {2}}}{(t^{2}+1)^{2}}}{{\frac {1-2{\sqrt {2}}t}{t^{2}+1}}{\frac {-{\sqrt {2}}t^{2}+t+{\sqrt {2}}}{t^{2}+1}}}}\;dt\\&=\int \!{\frac {-2}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\int {\frac {-2{\sqrt {2}}}{-2{\sqrt {2}}t+1}}dt\\&={\frac {1}{\sqrt {2}}}\ln {\Biggl |}2{\sqrt {2}}t-1{\Biggl |}+C\\&={\frac {\sqrt {2}}{2}}\ln {\Biggl |}2{\sqrt {2}}{\frac {{\sqrt {-x^{2}+x+2}}-{\sqrt {2}}}{x}}-1{\Biggl |}+C\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Tercera_sustitución_de_Euler"><span id="Tercera_sustituci.C3.B3n_de_Euler"></span>Tercera sustitución de Euler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=9" title="Editar sección: Tercera sustitución de Euler"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Para evaluar </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 \!{\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \!{\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/accc45df3da8c878b180948d6b1426637aca4c95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.654ex; height:7.009ex;" alt="{\displaystyle \int \!{\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx,}"></span></dd></dl> <p>se puede usar la tercera sustitución y configurar <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 {\sqrt {-(x-2)(x-1)}}=(x-2)t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-(x-2)(x-1)}}=(x-2)t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e269cb334eb23ed4cc52b83cebcea41aca93ba0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:29.495ex; height:4.843ex;" alt="{\displaystyle {\sqrt {-(x-2)(x-1)}}=(x-2)t}"></span>. Así </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 x={\frac {-2t^{2}-1}{-t^{2}-1}}\qquad \ dx={\frac {2t}{(-t^{2}-1)^{2}}}\,\ dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-2t^{2}-1}{-t^{2}-1}}\qquad \ dx={\frac {2t}{(-t^{2}-1)^{2}}}\,\ dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b78c7b129c360a901c35ba3cbf43270c413c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.076ex; height:6.509ex;" alt="{\displaystyle x={\frac {-2t^{2}-1}{-t^{2}-1}}\qquad \ dx={\frac {2t}{(-t^{2}-1)^{2}}}\,\ dt,}"></span></dd></dl> <p>y </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 {\sqrt {-x^{2}+3x-2}}=(x-2)t={\frac {t}{-t^{2}-1.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </msqrt> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x^{2}+3x-2}}=(x-2)t={\frac {t}{-t^{2}-1.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d100d27d4215003b0955d52f54deb5931839cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.217ex; height:5.676ex;" alt="{\displaystyle {\sqrt {-x^{2}+3x-2}}=(x-2)t={\frac {t}{-t^{2}-1.}}}"></span></dd></dl> <p>A continuación, </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 {\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx=\int {\frac {({\frac {-2t^{2}-1}{-t^{2}-1}})^{2}{\frac {2t}{(-t^{2}-1)^{2}}}}{\frac {t}{-t^{2}-1}}}\ dt=\int {\frac {2(-2t^{2}-1)^{2}}{(-t^{2}-1)^{3}}}\ dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> </msqrt> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mfrac> <mi>t</mi> <mrow> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx=\int {\frac {({\frac {-2t^{2}-1}{-t^{2}-1}})^{2}{\frac {2t}{(-t^{2}-1)^{2}}}}{\frac {t}{-t^{2}-1}}}\ dt=\int {\frac {2(-2t^{2}-1)^{2}}{(-t^{2}-1)^{3}}}\ dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc9202e8f2436e79309c3637fee4d5e9beec6bd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:71.315ex; height:9.843ex;" alt="{\displaystyle \int {\frac {x^{2}}{\sqrt {-x^{2}+3x-2}}}\ dx=\int {\frac {({\frac {-2t^{2}-1}{-t^{2}-1}})^{2}{\frac {2t}{(-t^{2}-1)^{2}}}}{\frac {t}{-t^{2}-1}}}\ dt=\int {\frac {2(-2t^{2}-1)^{2}}{(-t^{2}-1)^{3}}}\ dt.}"></span></dd></dl> <p>Como se puede ver, esta es una función racional que se puede resolver usando fracciones parciales. </p> <div class="mw-heading mw-heading2"><h2 id="Generalizaciones">Generalizaciones</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=10" title="Editar sección: Generalizaciones"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Las sustituciones de Euler se pueden generalizar permitiendo el uso de números imaginarios. Por ejemplo, en la integral <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 \textstyle \int {\frac {dx}{\sqrt {-x^{2}+c}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int {\frac {dx}{\sqrt {-x^{2}+c}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6067a9a69977fd5b4d4ebf0a331f03189d9c733b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.325ex; height:4.676ex;" alt="{\displaystyle \textstyle \int {\frac {dx}{\sqrt {-x^{2}+c}}}}"></span>, se puede usar la sustitución <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 {\sqrt {-x^{2}+c}}=\pm ix+t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mo>±</mo> <mi>i</mi> <mi>x</mi> <mo>+</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {-x^{2}+c}}=\pm ix+t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3d18c6fa99b74fcb8fc89877cd34e4e32bb6af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.082ex; height:3.509ex;" alt="{\displaystyle {\sqrt {-x^{2}+c}}=\pm ix+t}"></span>. Las extensiones a los números complejos permiten usar todo tipo de sustituciones de Euler independientemente de los coeficientes de la expresión cuadrática. </p><p>Las sustituciones de Euler se pueden generalizar a una clase más amplia de funciones. Considérense las integrales de la forma </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 R_{1}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}\,\log {\Big (}R_{2}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}{\Big )}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mi>x</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int R_{1}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}\,\log {\Big (}R_{2}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}{\Big )}\,dx,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e271c834837fdf033eb629a544bdd6bc5494452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.888ex; height:5.676ex;" alt="{\displaystyle \int R_{1}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}\,\log {\Big (}R_{2}{\Big (}x,{\sqrt {ax^{2}+bx+c}}{\Big )}{\Big )}\,dx,}"></span></dd></dl> <p>donde <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 R_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d63c96f59d98589d923c4f0b04222feaa7283e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{1}}"></span> y <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 R_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f571121c264178676d1df8ab899f238a39bc2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{2}}"></span> son funciones racionales de <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> y <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 {\sqrt {ax^{2}+bx+c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11567a8ebd2e19291b9c733b1781fed116483262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.952ex; height:3.509ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}}"></span>. Esta integral se puede transformar mediante la sustitución <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 {\sqrt {ax^{2}+bx+c}}={\sqrt {a}}+xt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>a</mi> </msqrt> </mrow> <mo>+</mo> <mi>x</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {ax^{2}+bx+c}}={\sqrt {a}}+xt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f4349ec0deca4d2d1c2d3b1cad6cc93d397e3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.226ex; height:3.676ex;" alt="{\displaystyle {\sqrt {ax^{2}+bx+c}}={\sqrt {a}}+xt}"></span> en otra integral </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 {\tilde {R}}_{1}(t)\log {\big (}{\tilde {R}}_{2}(t){\big )}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\tilde {R}}_{1}(t)\log {\big (}{\tilde {R}}_{2}(t){\big )}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d79352aff557a64be5e6d7cc6e2080f5908c2ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.48ex; height:5.676ex;" alt="{\displaystyle \int {\tilde {R}}_{1}(t)\log {\big (}{\tilde {R}}_{2}(t){\big )}\,dt,}"></span></dd></dl> <p>donde <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 {\tilde {R}}_{1}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {R}}_{1}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b9caefd2bd447440e6d85ee84879f4dab3e165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.467ex; height:3.176ex;" alt="{\displaystyle {\tilde {R}}_{1}(t)}"></span> y <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 {\tilde {R}}_{2}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {R}}_{2}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a9e66a54cf23ac8c2f35eac246064435d52e92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.467ex; height:3.176ex;" alt="{\displaystyle {\tilde {R}}_{2}(t)}"></span> ahora son simplemente funciones racionales de <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>. En principio, utilizando la <a href="/wiki/Factorizaci%C3%B3n" title="Factorización">factorización</a> y la <a href="/wiki/Descomposici%C3%B3n_en_fracciones_simples" title="Descomposición en fracciones simples">descomposición en fracciones simples</a> se puede dividir la integral en términos simples, que se pueden integrar analíticamente mediante el uso de la función <a href="/wiki/Dilogaritmo" class="mw-redirect" title="Dilogaritmo">dilogaritmo</a>.<sup id="cite_ref-2" class="reference separada"><a href="#cite_note-2"><span class="corchete-llamada">[</span>2<span class="corchete-llamada">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Véase_también"><span id="V.C3.A9ase_tambi.C3.A9n"></span>Véase también</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=11" title="Editar sección: Véase también"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Archivo:Nuvola_apps_edu_mathematics-p.svg" class="mw-file-description" title="Ver el portal sobre Matemáticas"><img alt="Ver el portal sobre Matemáticas" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/20px-Nuvola_apps_edu_mathematics-p.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/30px-Nuvola_apps_edu_mathematics-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Nuvola_apps_edu_mathematics-p.svg/40px-Nuvola_apps_edu_mathematics-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <a href="/wiki/Portal:Matem%C3%A1ticas" class="mw-redirect" title="Portal:Matemáticas">Portal:Matemáticas</a>. Contenido relacionado con <b><a href="/wiki/Matem%C3%A1ticas" title="Matemáticas">Matemáticas</a></b>.</li> <li><a href="/wiki/M%C3%A9todos_de_integraci%C3%B3n" title="Métodos de integración">Métodos de integración</a></li> <li><a href="/wiki/Sustituci%C3%B3n_trigonom%C3%A9trica" class="mw-redirect" title="Sustitución trigonométrica">Sustitución trigonométrica</a></li> <li><a href="/wiki/Sustituci%C3%B3n_de_Weierstrass" title="Sustitución de Weierstrass">Sustitución de Weierstrass</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referencias">Referencias</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=12" title="Editar sección: Referencias"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="listaref" style="list-style-type: decimal;"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">N. Piskunov, <i>Diferentsiaal- ja integraalarvutus körgematele tehnilistele öppeasutustele. Viies, taiendatud trukk. Kirjastus Valgus</i>, Tallinn (1965). Note: Euler substitutions can be found in most Russian calculus textbooks.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><span id="CITAREFZwillinger" class="citation libro">Zwillinger, Daniel. <i>The Handbook of Integration</i>. 1992: Jones and Bartlett. pp. 145-146. <small><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/Especial:FuentesDeLibros/978-0867202939" title="Especial:FuentesDeLibros/978-0867202939">978-0867202939</a></small>.</span><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fes.wikipedia.org%3ASustituci%C3%B3n+de+Euler&rft.au=Zwillinger%2C+Daniel&rft.aufirst=Daniel&rft.aulast=Zwillinger&rft.btitle=The+Handbook+of+Integration&rft.genre=book&rft.isbn=978-0867202939&rft.pages=145-146&rft.place=1992&rft.pub=Jones+and+Bartlett&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Enlaces_externos">Enlaces externos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Sustituci%C3%B3n_de_Euler&action=edit&section=13" title="Editar sección: Enlaces externos"><span>editar</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i>Este artículo incorpora material de <a href="https://planetmath.org/alphabetical.html" class="extiw" title="planetmath:39681">Eulers Substitutions For Integration</a> en <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, que tiene licencia <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/2.5/">Creative Commons Atribución Compartir-Igual</a>.</i></li></ul> <style data-mw-deduplicate="TemplateStyles:r161257576">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox table{margin:0}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control .mw-mf-linked-projects{display:flex;padding:0.5em;border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral,#eaecf0);color:var(--color-base,#202122)}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}.mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#a2a9b1);background-color:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .mw-authority-control .navbox-list{border-color:#f8f9fa}.mw-parser-output .mw-authority-control .navbox th{background-color:#eeeeff}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d);background-color:var(--background-color-neutral,#27292d);color:var(--color-base,#eaecf0)}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-night .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .mw-mf-linked-projects{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral,#27292d)!important;color:var(--color-base,#eaecf0)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox{border:1px solid var(--border-color-base,#72777d)!important;background-color:var(--background-color-neutral-subtle,#202122)!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox-list{border-color:#202122!important}html.skin-theme-clientpref-os .mw-parser-output .mw-authority-control .navbox th{background-color:#27292d!important}}</style><div class="mw-authority-control"><div role="navigation" class="navbox" aria-label="Navbox" style="width: inherit;padding:3px"><table class="hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width: 12%; 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