GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Integration durch Substitution</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading">Integration durch Substitution</h1>

</header>

</div>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">Die Integration durch Substitution oder die Substitutionsregel ist eine wichtige Methode in der <a href="Integralrechnung" title="Integralrechnung">Integralrechnung</a>, um <a href="Stammfunktion" title="Stammfunktion">Stammfunktionen</a> zu finden und <a href="Bestimmtes_Integral" class="mw-redirect" title="Bestimmtes Integral">bestimmte Integrale</a> auszuwerten. Die Substitutionsmethode erlaubt es, einen „komplizierten“ <a href="Integralrechnung#Bezeichnungen" title="Integralrechnung">Integranden</a> durch einen „einfachen“ Integranden zu ersetzen und damit das gegebene Integral auf ein einfacher handhabbares Integral zurückzuführen. Der im Hintergrund der Substitutionsmethode stehende <a href="Transformationssatz" title="Transformationssatz">Transformationssatz</a> gehört zu den wichtigsten Sätzen der <a href="Analysis" title="Analysis">Analysis</a>.

Die Substitutionsregel der <a href="Integralrechnung" title="Integralrechnung">Integralrechnung</a> ist die Umkehrung der <a href="Kettenregel" title="Kettenregel">Kettenregel</a> der <a href="Differentialrechnung" title="Differentialrechnung">Differentialrechnung</a>. Bei Integralen über Funktionen mehrerer Variablen kommt der <a href="Transformationssatz" title="Transformationssatz">Transformationssatz</a> zur Anwendung, der allerdings eine <a href="Bijektivit%C3%A4t" class="mw-redirect" title="Bijektivität">bijektive</a> Substitutionsfunktion verlangt.

<div class="mw-heading mw-heading2"><h2 id="Aussage_der_Substitutionsregel">Aussage der Substitutionsregel</h2></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon I\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon I\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa91b80587ee863c353413b3919b8704ef799018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.776ex; height:2.509ex;" alt="{\displaystyle f\colon I\to \mathbb {R} }" loading="lazy"> eine <a href="Stetige_Funktion" title="Stetige Funktion">stetige</a> Funktion auf einem reellen <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon [a,b]\to I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon [a,b]\to I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1323ec099c6d2d9a8037b76d044320b278834ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.895ex; height:2.843ex;" alt="{\displaystyle \varphi \colon [a,b]\to I}" loading="lazy"> eine <a href="Stetig_differenzierbare_Funktion" class="mw-redirect" title="Stetig differenzierbare Funktion">stetig differenzierbare</a> Funktion, so gilt

</mrow>

</mrow>

</munderover>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad1bf944456a63718d28dc345956de2ea16e872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.667ex; height:10.009ex;" alt="{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}" loading="lazy"><a href="#cite_note-1">[1]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Heuristische_Herleitung">Heuristische Herleitung</h2></div>

Die Substitutionsregel lässt sich mithilfe des Differentialkalküls herleiten: Dazu substituiert man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\varphi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\varphi (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b35ef1efa72359945035fa9aecf2fecde115344a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.597ex; height:2.843ex;" alt="{\displaystyle t=\varphi (x)}" loading="lazy"> und schreibt die Ableitung als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {\mathrm {d} t}{\mathrm {d} x}}=\varphi '(x)}">

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {\mathrm {d} t}{\mathrm {d} x}}=\varphi '(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1f79e8846c4fb52c56e0a7dc9aa132a6df1a779.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.133ex; height:3.843ex;" alt="{\displaystyle {\tfrac {\mathrm {d} t}{\mathrm {d} x}}=\varphi '(x)}" loading="lazy">. Die linke Seite dieser Gleichung fasst man als Quotient von zwei <a href="Differential_(Mathematik)" title="Differential (Mathematik)">Differentialen</a> auf, wodurch man nach Multiplikation mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b637d7a6d64d797391d40ceae9e8696e5d76f15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.622ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x}" loading="lazy"> die Gleichung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi '(x)\,\mathrm {d} x=\mathrm {d} t}">

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi '(x)\,\mathrm {d} x=\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d345efc931714ced992aa5b11611706f43f2211.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.584ex; height:3.009ex;" alt="{\displaystyle \varphi '(x)\,\mathrm {d} x=\mathrm {d} t}" loading="lazy"> erhält. Durch Einsetzen in das Integral erhält man

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int f(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ffa6c7162e9d6671c50fd4b4567e8d904dba56b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.133ex; height:5.676ex;" alt="{\displaystyle \int f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int f(t)\,\mathrm {d} t.}" loading="lazy"></dd></dl>

Im linken Integral ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> die Integrationsvariable, im rechten Integral nun <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy">.

Bei bestimmten Integralen erfordert dieser Wechsel der Integrationsvariablen noch eine Anpassung der Integrationsgrenzen: Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\varphi (a)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\varphi (a)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f75c3ed254073950103ab6fcf82e1973c85431be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.497ex; height:2.843ex;" alt="{\displaystyle t=\varphi (a)}" loading="lazy"> und für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229dd3f0f42d375fa257bdc1389f92f7b225c415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\varphi (b)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\varphi (b)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d61924d1118faaac2fba07e80f0718190e6c32b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.265ex; height:2.843ex;" alt="{\displaystyle t=\varphi (b)}" loading="lazy">. Damit erhält man schließlich

</mrow>

</mrow>

</munderover>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</munderover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad1bf944456a63718d28dc345956de2ea16e872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.667ex; height:10.009ex;" alt="{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Beweis">Beweis</h2></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine <a href="Stammfunktion" title="Stammfunktion">Stammfunktion</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, so gilt für die Ableitung der <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">zusammengesetzten Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\circ \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\circ \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/530441dcc92e7254eae888be43d3a6d6954b3cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.456ex; height:2.676ex;" alt="{\displaystyle F\circ \varphi }" loading="lazy"> nach der <a href="Kettenregel" title="Kettenregel">Kettenregel</a>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (F\circ \varphi )'(x)=F'(\varphi (x))\cdot \varphi '(x)=f(\varphi (x))\cdot \varphi '(x).}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c9b5e09dd32d45a1bc8e0eb0a109135c31dd48e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.693ex; height:3.009ex;" alt="{\displaystyle (F\circ \varphi )'(x)=F'(\varphi (x))\cdot \varphi '(x)=f(\varphi (x))\cdot \varphi '(x).}" loading="lazy"></dd></dl>

Also ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\circ \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\circ \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/530441dcc92e7254eae888be43d3a6d6954b3cdb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.456ex; height:2.676ex;" alt="{\displaystyle F\circ \varphi }" loading="lazy"> eine Stammfunktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\circ \varphi )\cdot \varphi '}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\circ \varphi )\cdot \varphi '}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/156c4d6ec5bc58307b106761d298683fa479a653.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.687ex; height:3.009ex;" alt="{\displaystyle (f\circ \varphi )\cdot \varphi '}" loading="lazy">.

Durch zweimaliges Anwenden des <a href="Hauptsatz_der_Differential-_und_Integralrechnung" class="mw-redirect" title="Hauptsatz der Differential- und Integralrechnung">Hauptsatzes der Differential- und Integralrechnung</a> erhält man die Substitutionsregel:

</mrow>

</mrow>

</munderover>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=(F\circ \varphi )(x){\bigg \vert }_{a}^{b}=F(\varphi (b))-F(\varphi (a))=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d6210683da9f13d22541b54721ece224fe6882b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:71.843ex; height:10.009ex;" alt="{\displaystyle \int \limits _{a}^{b}f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x=(F\circ \varphi )(x){\bigg \vert }_{a}^{b}=F(\varphi (b))-F(\varphi (a))=\int \limits _{\varphi (a)}^{\varphi (b)}f(t)\,\mathrm {d} t.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Transformationssatz">Transformationssatz</h2></div>

Die Substitutionsmethode lässt sich unter etwas engeren Voraussetzungen auch „rückwärts“ durchführen. Das ist die Substitution 2. Art. Ausgangspunkt ist für eine stetige Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon I\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon I\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa91b80587ee863c353413b3919b8704ef799018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.776ex; height:2.509ex;" alt="{\displaystyle f\colon I\to \mathbb {R} }" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha ,\beta \in I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha ,\beta \in I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3e56cfc7cb92a2183ee3896eebb0f296b0c98dbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle \alpha ,\beta \in I}" loading="lazy"> das Integral

<mtr>

<mtd>

</mrow>

</mrow>

</munderover>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int \limits _{\alpha }^{\beta }f(x)\,\mathrm {d} x.\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5651ce2fd78f153cd118340d076b5dba80d48437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:11.406ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\int \limits _{\alpha }^{\beta }f(x)\,\mathrm {d} x.\end{aligned}}}" loading="lazy"></dd></dl>

Man benutzt eine Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon [a,b]\to I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon [a,b]\to I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1323ec099c6d2d9a8037b76d044320b278834ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.895ex; height:2.843ex;" alt="{\displaystyle \varphi \colon [a,b]\to I}" loading="lazy">, die <a href="Injektive_Funktion" title="Injektive Funktion">injektiv</a> und stetig differenzierbar ist. Dann existiert die Umkehrfunktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi ^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5393676141f43c07be34b09aeb65f22064e92cf7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.853ex; height:3.176ex;" alt="{\displaystyle \varphi ^{-1}}" loading="lazy">. Man kann die Substitutionsregel nun von rechts nach links lesen:

</mrow>

</mrow>

</munderover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munderover>

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \limits _{\alpha }^{\beta }f(x)\,\mathrm {d} x=\int \limits _{\varphi ^{-1}(\alpha )}^{\varphi ^{-1}(\beta )}f(\varphi (t))\varphi '(t)\,\mathrm {d} t.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8510713a725ce1bb49e6cb85f949d257d8e494a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:34.577ex; height:10.676ex;" alt="{\displaystyle \int \limits _{\alpha }^{\beta }f(x)\,\mathrm {d} x=\int \limits _{\varphi ^{-1}(\alpha )}^{\varphi ^{-1}(\beta )}f(\varphi (t))\varphi '(t)\,\mathrm {d} t.}" loading="lazy"></dd></dl>

Sie lässt sich wie folgt interpretieren: Transformiert man die Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> mittel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\varphi (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=\varphi (t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b55af9961104e5de0efdd347c0c12317ed84d76d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.597ex; height:2.843ex;" alt="{\displaystyle x=\varphi (t)}" loading="lazy">, so ändert sich der Wert des Integrals nicht, wenn man die neue Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be3cbb92d31c8450e3c2775422a68d706e80f774.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.993ex; height:2.676ex;" alt="{\displaystyle f\circ \varphi }" loading="lazy"> mit der Ableitung von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> multipliziert und die Integralgrenzen wie oben anpasst.<a href="#cite_note-2">[2]</a> In dieser Fassung nennt man die Substitutionsregel deshalb auch Transformationsformel.<a href="#cite_note-3">[3]</a>

Bei geschickter Wahl der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"> kann entgegen dem ersten Anschein der Integrand vereinfacht werden.

<div class="mw-heading mw-heading2"><h2 id="Substitution_eines_bestimmten_Integrals">Substitution eines bestimmten Integrals</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_1">Beispiel 1</h3></div>

Berechnung des Integrals

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{a}\sin(2x)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/765daacfc44ae3dc09cf78e573f192892712cb46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.119ex; height:5.843ex;" alt="{\displaystyle \int _{0}^{a}\sin(2x)\,\mathrm {d} x}" loading="lazy"></dd></dl>

für eine beliebige reelle Zahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a>0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f34a80ea013edb56e340b19550430a8b6dfd7b9.svg" 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}" loading="lazy">:

Durch die Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\varphi (x)=2x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\varphi (x)=2x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98f9db8b4cef37b529921b0e4b0a1e7b448e963e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.188ex; height:2.843ex;" alt="{\displaystyle t=\varphi (x)=2x}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=\varphi '(x)\,\mathrm {d} x=2\,\mathrm {d} x}">

</mrow>

<msup>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=\varphi '(x)\,\mathrm {d} x=2\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c1afb2e9ac8e67959b102cd70a5c582f9d4ef167.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.854ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} t=\varphi '(x)\,\mathrm {d} x=2\,\mathrm {d} x}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x={\tfrac {1}{2}}{\mathrm {d} t}}">

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x={\tfrac {1}{2}}{\mathrm {d} t}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a7e849992e3dca6efbc1a2c7d8e206f19b40769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.511ex; height:3.509ex;" alt="{\displaystyle \mathrm {d} x={\tfrac {1}{2}}{\mathrm {d} t}}" loading="lazy">, und damit:

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{a}\sin(2x)\,\mathrm {d} x=\int _{\varphi (0)}^{\varphi (a)}\sin(t)\,{\frac {1}{2}}{\mathrm {d} t}=\int _{0}^{2a}\sin(t)\,{\frac {1}{2}}{\mathrm {d} t}={\frac {1}{2}}\int _{0}^{2a}\sin(t)\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a51bbd7428e2505b2e5f468498a44ce346cd3702.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:69.727ex; height:6.676ex;" alt="{\displaystyle \int _{0}^{a}\sin(2x)\,\mathrm {d} x=\int _{\varphi (0)}^{\varphi (a)}\sin(t)\,{\frac {1}{2}}{\mathrm {d} t}=\int _{0}^{2a}\sin(t)\,{\frac {1}{2}}{\mathrm {d} t}={\frac {1}{2}}\int _{0}^{2a}\sin(t)\,\mathrm {d} t}" loading="lazy"></dd>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{2}}[-\cos(t)]_{0}^{2a}={\frac {1}{2}}((-\cos(2a))-(-\cos(0)))={\frac {1}{2}}(-\cos(2a)+1)={\frac {1}{2}}(1-\cos(2a))}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7d75cc65eaa46d3bb5e89455098a94b015e31c3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:85.415ex; height:5.176ex;" alt="{\displaystyle ={\frac {1}{2}}[-\cos(t)]_{0}^{2a}={\frac {1}{2}}((-\cos(2a))-(-\cos(0)))={\frac {1}{2}}(-\cos(2a)+1)={\frac {1}{2}}(1-\cos(2a))}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_2">Beispiel 2</h3></div>

Berechnung des Integrals

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2}x\cos \left(x^{2}+1\right)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a3196a0d35cab1e8d82cbb453895b53c28af5c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.259ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{2}x\cos \left(x^{2}+1\right)\,\mathrm {d} x}" loading="lazy">:</dd></dl>

Durch die Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\varphi (x)=x^{2}+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\varphi (x)=x^{2}+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae25e9a12eb4bb6e919a344244e01d55380cf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.082ex; height:3.176ex;" alt="{\displaystyle t=\varphi (x)=x^{2}+1}" loading="lazy"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b1a8091cb1d39ee76e4da86274119c2e85a9f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.732ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}" loading="lazy">, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,\mathrm {d} x={\tfrac {1}{2}}\mathrm {d} t}">

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\,\mathrm {d} x={\tfrac {1}{2}}\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72d39b3e072fc8e9cf03d3210fdf23e601cde3d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.228ex; height:3.509ex;" alt="{\displaystyle x\,\mathrm {d} x={\tfrac {1}{2}}\mathrm {d} t}" loading="lazy">, und damit

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2}x\cos \left(x^{2}+1\right)\,\mathrm {d} x={\frac {1}{2}}\int _{1}^{5}\cos(t)\,\mathrm {d} t={\frac {1}{2}}\left(\sin(5)-\sin(1)\right)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb9f4617db0cd8e8c921c90f8893a9c679bdf490.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.717ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{2}x\cos \left(x^{2}+1\right)\,\mathrm {d} x={\frac {1}{2}}\int _{1}^{5}\cos(t)\,\mathrm {d} t={\frac {1}{2}}\left(\sin(5)-\sin(1)\right)}" loading="lazy">.</dd></dl>

Es wird also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92a3a8d23f9f8123651e496dcf8490990c65cf9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"> ersetzt und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\,\mathrm {d} x}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/406c9933c202af9b7d77255155dfa9ba6c7901f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.339ex; height:2.176ex;" alt="{\displaystyle x\,\mathrm {d} x}" loading="lazy"> durch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\,\mathrm {d} t}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/918f3a36de17f963461dd35a7ed372492399f39d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.177ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\,\mathrm {d} t}" loading="lazy">.

Die untere Grenze des Integrals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}" loading="lazy"> wird dabei in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(0)=0^{2}+1=1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t(0)=0^{2}+1=1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e975db584e7acf05c600aea1bb93c07e6015bdf1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.39ex; height:3.176ex;" alt="{\displaystyle t(0)=0^{2}+1=1}" loading="lazy"> umgewandelt und die obere Grenze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=2}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f39b6e42e5ffb81ac7b051b9e48b9a91d0713c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=2}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t(2)=2^{2}+1=5}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t(2)=2^{2}+1=5}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e857a75e63cf62df7d570778c8d0500c0f77b2a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.39ex; height:3.176ex;" alt="{\displaystyle t(2)=2^{2}+1=5}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Beispiel_3">Beispiel 3</h3></div>

Das ist ein Beispiel für die Substitution rückwärts (Substitution 2. Art).

Für die Berechnung des Integrals

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1eeb3e2fd8fb703172a867370c78e4c8bc4cba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.645ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t}" loading="lazy"></dd></dl>

kann man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\sin(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\sin(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f94054ea1301b2d3d32ba4bfad05bbad5846278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.933ex; height:2.843ex;" alt="{\displaystyle t=\sin(x)}" loading="lazy"> substituieren (eine <a href="Weierstra%C3%9F-Substitution" title="Weierstraß-Substitution">Weierstraß-Substitution</a>). Daraus ergibt sich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca38e010c0d65d593ee571d3c26ae846a198d95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.49ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}" loading="lazy">. Um die Integrationsgrenzen umzurechnen, benutzt man die umgekehrte Beziehung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=\arcsin(t)}">

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=\arcsin(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/206f739c0b7b322ed049c336f4067ac78e5d5a47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.039ex; height:2.843ex;" alt="{\displaystyle x=\arcsin(t)}" loading="lazy">. Die obere Grenze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> wird zu <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\pi }{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98f98bef5d4981ff6e2aa827d4699e347fb30db2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:2.168ex; height:4.676ex;" alt="{\displaystyle {\frac {\pi }{2}}}" loading="lazy">, weil <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(1)={\frac {\pi }{2}}}">

<mi>arcsin</mi>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin(1)={\frac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03d5b8ec9d72063a9c8837b2f533a9d019845ddc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.2ex; height:4.676ex;" alt="{\displaystyle \arcsin(1)={\frac {\pi }{2}}}" loading="lazy">. Aus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \arcsin(0)=0}">

<mi>arcsin</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \arcsin(0)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b60d669653eb991a58122581a70bfcb5972faa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.195ex; height:2.843ex;" alt="{\displaystyle \arcsin(0)=0}" loading="lazy"> ergibt sich die neue untere Grenze <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">.

Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {1-\sin ^{2}(t)}}=\cos(t)}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {1-\sin ^{2}(t)}}=\cos(t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8c119758b553bf604ea9ff892dd60e32c475960d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:21.744ex; height:4.843ex;" alt="{\displaystyle {\sqrt {1-\sin ^{2}(t)}}=\cos(t)}" loading="lazy"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq t\leq {\frac {\pi }{2}}}">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq t\leq {\frac {\pi }{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f92bfe1511bd87bc2693c7115344ab3adf2dee7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.367ex; height:4.676ex;" alt="{\displaystyle 0\leq t\leq {\frac {\pi }{2}}}" loading="lazy"> rechnet man

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msubsup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-\sin ^{2}(x)}}\cos(x)\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}\cos(x)\cdot \cos(x)\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}\cos ^{2}(x)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e92b91e7e0fa7fa58f6637f611a02bcb2787340.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:88.518ex; height:6.509ex;" alt="{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t=\int _{0}^{\frac {\pi }{2}}{\sqrt {1-\sin ^{2}(x)}}\cos(x)\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}\cos(x)\cdot \cos(x)\,\mathrm {d} x=\int _{0}^{\frac {\pi }{2}}\cos ^{2}(x)\,\mathrm {d} x}" loading="lazy">.</dd></dl>

Das Integral in der letzten Zeile kann mit <a href="Partielle_Integration" title="Partielle Integration">partieller Integration</a> oder mit der <a href="Formelsammlung_Trigonometrie#Kosinus" title="Formelsammlung Trigonometrie">trigonometrischen Formel</a>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75b8619e312258523d7c6b6f11afda26b6d82dc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.654ex; height:5.676ex;" alt="{\displaystyle \cos ^{2}(x)={\frac {1+\cos(2x)}{2}}}" loading="lazy"></dd></dl>

und einer weiteren Substitution berechnet werden. Es ergibt sich

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t=\left[{\frac {x}{2}}+{\frac {1}{4}}\sin(2x)\right]_{x=0}^{x={\frac {\pi }{2}}}={\frac {\pi }{4}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87c2e82ea7c116d7c0c94461f81e31266313eac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.066ex; height:7.009ex;" alt="{\displaystyle \int _{0}^{1}{\sqrt {1-t^{2}}}\,\mathrm {d} t=\left[{\frac {x}{2}}+{\frac {1}{4}}\sin(2x)\right]_{x=0}^{x={\frac {\pi }{2}}}={\frac {\pi }{4}}}" loading="lazy">.</dd></dl>

(Damit haben wir die Fläche eines Viertelkreises berechnet.)

<div class="mw-heading mw-heading2"><h2 id="Substitution_eines_unbestimmten_Integrals">Substitution eines unbestimmten Integrals</h2></div>

Hingewiesen sei auf die Problematik des Begriffs „<a href="Stammfunktion" title="Stammfunktion">unbestimmtes Integral</a>“, insbesondere in der Notation.

<div class="mw-heading mw-heading3"><h3 id="Voraussetzungen_und_Vorgehen">Voraussetzungen und Vorgehen</h3></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon I\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon I\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa91b80587ee863c353413b3919b8704ef799018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.776ex; height:2.509ex;" alt="{\displaystyle f\colon I\to \mathbb {R} }" loading="lazy"> eine <a href="Stetige_Funktion" title="Stetige Funktion">stetige</a> Funktion auf einem reellen <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \colon [a,b]\to I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi \colon [a,b]\to I}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1323ec099c6d2d9a8037b76d044320b278834ff6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.895ex; height:2.843ex;" alt="{\displaystyle \varphi \colon [a,b]\to I}" loading="lazy"> eine <a href="Stetig_differenzierbare_Funktion" class="mw-redirect" title="Stetig differenzierbare Funktion">stetig differenzierbare</a> Funktion, so gilt

<msup>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(\varphi (x))=\int f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/833fb5847878674c3b8a81a8f8805b1d5d3a0168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.315ex; height:5.676ex;" alt="{\displaystyle F(\varphi (x))=\int f(\varphi (x))\cdot \varphi '(x)\,\mathrm {d} x.}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine Stammfunktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist.

Das Entscheidende bei der Substitution in einem unbestimmten Integral ist, dass am Ende der Rechnung die substituierte Variable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" 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}" loading="lazy"> wieder durch den Term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4c4046f1f2de7df04bde418ba2bc4d3898ac2385.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.659ex; height:2.843ex;" alt="{\displaystyle \varphi (x)}" loading="lazy"> ersetzt werden muss (Rücksubstitution).

<div class="mw-heading mw-heading3"><h3 id="Beispiel_1_2">Beispiel 1</h3></div>

Durch <a href="Quadratische_Erg%C3%A4nzung" title="Quadratische Ergänzung">quadratische Ergänzung</a> und anschließende Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=x+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=x+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f37774a96efc923c52a17c57e2dfa9340ea7d316.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.271ex; height:2.343ex;" alt="{\displaystyle t=x+1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} x=\mathrm {d} t}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x=\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/439a133ba5960ff74b9326c31ae39a8a5dd09c7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.853ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} x=\mathrm {d} t}" loading="lazy"> erhält man

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mi>arctan</mi>

<mi>arctan</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {1}{x^{2}+2x+2}}\,\mathrm {d} x=\int {\frac {1}{(x+1)^{2}+1}}\,\mathrm {d} x=\int {\frac {1}{t^{2}+1}}\,\mathrm {d} t=\arctan(t)+C=\arctan(x+1)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0fd5878c477b0b81ee5c4d138354604664bc269e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:92.934ex; height:6.009ex;" alt="{\displaystyle \int {\frac {1}{x^{2}+2x+2}}\,\mathrm {d} x=\int {\frac {1}{(x+1)^{2}+1}}\,\mathrm {d} x=\int {\frac {1}{t^{2}+1}}\,\mathrm {d} t=\arctan(t)+C=\arctan(x+1)+C}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_2_2">Beispiel 2</h3></div>

Mit der Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=x^{2}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3712979baadd13ed75826ec1c6d74676096e7958.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.322ex; height:2.676ex;" alt="{\displaystyle t=x^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/55b1a8091cb1d39ee76e4da86274119c2e85a9f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.732ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} t=2x\,\mathrm {d} x}" loading="lazy"> erhält man

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int x\,\cos \left(x^{2}\right)\,\mathrm {d} x={\frac {1}{2}}\int 2x\cos \left(x^{2}\right)\,\mathrm {d} x={\frac {1}{2}}\int \cos(t)\,\mathrm {d} t={\frac {1}{2}}\left(\sin(t)+C'\right)={\frac {1}{2}}\sin \left(x^{2}\right)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b76eff3e418aa76f998064ed3e6c0e08df7e5bfd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:88.822ex; height:5.676ex;" alt="{\displaystyle \int x\,\cos \left(x^{2}\right)\,\mathrm {d} x={\frac {1}{2}}\int 2x\cos \left(x^{2}\right)\,\mathrm {d} x={\frac {1}{2}}\int \cos(t)\,\mathrm {d} t={\frac {1}{2}}\left(\sin(t)+C'\right)={\frac {1}{2}}\sin \left(x^{2}\right)+C}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_3_2">Beispiel 3</h3></div>

Mit der Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\cos(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\cos(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2fcdc1d409a9e40680508ecd59d7365d5088dd1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.188ex; height:2.843ex;" alt="{\displaystyle t=\cos(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=-\sin(x)\,\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=-\sin(x)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ee8225e65a57c5bcd35683f1047146ac97c273b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.43ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} t=-\sin(x)\,\mathrm {d} x}" loading="lazy"> erhält man eine Stammfunktion für den <a href="Tangens_und_Kotangens" title="Tangens und Kotangens">Tangens</a>:

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \tan(x)\,\mathrm {d} x=\int -{\frac {-\sin(x)}{\cos(x)}}\,\mathrm {d} x=\int -{\frac {1}{t}}\,\mathrm {d} t=-\ln(|t|)+C=-\ln(|\cos(x)|)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/100687cf453c7547e05d195e0af093f18428a2fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:80.979ex; height:6.509ex;" alt="{\displaystyle \int \tan(x)\,\mathrm {d} x=\int -{\frac {-\sin(x)}{\cos(x)}}\,\mathrm {d} x=\int -{\frac {1}{t}}\,\mathrm {d} t=-\ln(|t|)+C=-\ln(|\cos(x)|)+C}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiel_4">Beispiel 4</h3></div>

Mit der Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=\sin(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=\sin(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f94054ea1301b2d3d32ba4bfad05bbad5846278.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.933ex; height:2.843ex;" alt="{\displaystyle t=\sin(x)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca38e010c0d65d593ee571d3c26ae846a198d95e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.49ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} t=\cos(x)\,\mathrm {d} x}" loading="lazy"> erhält man eine Stammfunktion für den <a href="Tangens_und_Kotangens" title="Tangens und Kotangens">Kotangens</a>:

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int \cot(x)\,\mathrm {d} x=\int {\frac {\cos(x)}{\sin(x)}}\,\mathrm {d} x=\int {\frac {1}{t}}\,\mathrm {d} t=\ln(|t|)+C=\ln(|\sin(x)|)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aabd8dc2c2ba888932e7a21b5790da6c970ae367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.517ex; height:6.509ex;" alt="{\displaystyle \int \cot(x)\,\mathrm {d} x=\int {\frac {\cos(x)}{\sin(x)}}\,\mathrm {d} x=\int {\frac {1}{t}}\,\mathrm {d} t=\ln(|t|)+C=\ln(|\sin(x)|)+C}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Spezialfälle_der_Substitution">Spezialfälle der Substitution</h2></div>

Nachfolgend wird davon ausgegangen, dass die Integrationsvariable mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> benannt ist.

<div class="mw-heading mw-heading3"><h3 id="Lineare_Substitution">Lineare Substitution</h3></div>

Erscheint in einem Integranden die Integrationsvariable <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" 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}" loading="lazy"> stets nur innerhalb eines Terms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax+b}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8507be931cbdb80c0b7339082b03eb0299710ae8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.397ex; height:2.343ex;" alt="{\displaystyle ax+b}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f455a7f96d74aa94573d8e32da3b240ab0aa294f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" loading="lazy">, so kann wie folgt vorgegangen werden: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> eine Stammfunktion von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, dann gilt

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int f(ax+b)\,\mathrm {d} x={\frac {1}{a}}F(ax+b)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bdecf475be161823a155620510f16631e0f17589.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.794ex; height:5.676ex;" alt="{\displaystyle \int f(ax+b)\,\mathrm {d} x={\frac {1}{a}}F(ax+b)+C}" loading="lazy">.</dd></dl>

Zum Beispiel gilt

<msup>

</mrow>

</msup>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int e^{3x+1}\,\mathrm {d} x={\frac {1}{3}}e^{(3x+1)}+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d77116782c9b03362ca8643f025a28531e6c91fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.93ex; height:5.676ex;" alt="{\displaystyle \int e^{3x+1}\,\mathrm {d} x={\frac {1}{3}}e^{(3x+1)}+C}" loading="lazy">,</dd></dl>

da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=e^{x}=F(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=e^{x}=F(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d38f7afd01857d642af2ae7920db9485d936cf2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.75ex; height:2.843ex;" alt="{\displaystyle f(x)=e^{x}=F(x)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a=3}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a0be4b8203e824561fa22ea831a2d86f2f028c9.svg" 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=3}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Logarithmische_Integration">Logarithmische Integration</h3></div>

Ist der Integrand ein Bruch, dessen Zähler die Ableitung des Nenners ist, kann das betreffende Integral schnell gelöst werden:

<mfrac>

<mrow>

<msup>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {f'(x)}{f(x)}}\,\mathrm {d} x=\ln |f(x)|+C.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2a9931b91e1f95bbf0c36ec4e74fda7c81d6e5ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.96ex; height:6.509ex;" alt="{\displaystyle \int {\frac {f'(x)}{f(x)}}\,\mathrm {d} x=\ln |f(x)|+C.}" loading="lazy"></dd></dl>

Es liegt hier eine Substitution 1. Art mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=f(x)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9adede852bec5c31fea9515d4787216c8ddb879a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.356ex; height:2.843ex;" alt="{\displaystyle t=f(x)}" loading="lazy"> vor.

Zum Beispiel gilt

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {x}{x^{2}+1}}\,\mathrm {d} x={\frac {1}{2}}\int {\frac {2x}{x^{2}+1}}\,\mathrm {d} x={\frac {1}{2}}\ln(x^{2}+1)+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/788e60f5b0eddfbbd283c166012d12de3698b2e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.336ex; height:5.676ex;" alt="{\displaystyle \int {\frac {x}{x^{2}+1}}\,\mathrm {d} x={\frac {1}{2}}\int {\frac {2x}{x^{2}+1}}\,\mathrm {d} x={\frac {1}{2}}\ln(x^{2}+1)+C}" loading="lazy">,</dd></dl>

da <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=x^{2}+1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=x^{2}+1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/060789c792de89c492dc74c0f3e74edd0cbf87c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.903ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{2}+1}" loading="lazy"> die Ableitung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'(x)=2x}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f'(x)=2x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c303b9422584925f8de55465ff05d23945903d4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.735ex; height:3.009ex;" alt="{\displaystyle f'(x)=2x}" loading="lazy"> hat.

<div class="mw-heading mw-heading3"><h3 id="Eulersche_Substitution">Eulersche Substitution</h3></div>

Nach einem Satz von Bernoulli lassen sich alle Integrale des Typs

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\;\mathrm {d} x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/999195605e6303223874b705257005dc35a4bfd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.801ex; height:5.676ex;" alt="{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\;\mathrm {d} x}" loading="lazy"></dd></dl>

und

<mfrac>

<mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d6369d5d875c9fc2c583895d3f1360001a1bb522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.369ex; height:6.676ex;" alt="{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {ax^{2}+bx+c}}}}" loading="lazy"></dd></dl>

elementar integrieren.

Euler hat hierzu mehrere Substitutionen 2. Art vorgeschlagen, die sich darin unterscheiden, welche Eigenschaften das konkrete Polynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/126c6935d3dd9f1c1da0c388ca2799be4f6f237c.svg" 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}" loading="lazy"> hat.

Beispiel:

<mfrac>

<mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}+1}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f2d919501a75bdca48622ad1c852bde6b4f032f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.127ex; height:6.676ex;" alt="{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}+1}}}}" loading="lazy"></dd></dl>

Die Substitution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=x+{\sqrt {x^{2}+1}}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=x+{\sqrt {x^{2}+1}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11dd5458b42b4df093679c3360701140a791bd1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.819ex; height:3.509ex;" alt="{\displaystyle t=x+{\sqrt {x^{2}+1}}}" loading="lazy"> führt zu

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x={\tfrac {t}{2}}-{\tfrac {1}{2t}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8cf803fb82b9a908102918f73c5138efafe1fa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.178ex; height:3.509ex;" alt="{\displaystyle x={\tfrac {t}{2}}-{\tfrac {1}{2t}}}" loading="lazy"> und

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} x=\left({\tfrac {1}{2}}+{\tfrac {1}{2t^{2}}}\right)\mathrm {d} t}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d423255c09856709c601bb2d4d6741bd9ce73cc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.597ex; height:4.843ex;" alt="{\displaystyle \mathrm {d} x=\left({\tfrac {1}{2}}+{\tfrac {1}{2t^{2}}}\right)\mathrm {d} t}" loading="lazy">.

Damit ergibt sich

<mfrac>

<mrow>

</mrow>

</mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mfrac>

</mrow>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<mrow>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}+1}}}=\int {\frac {{\frac {1}{2}}+{\frac {1}{2t^{2}}}}{{\frac {t}{2}}+{\frac {1}{2t}}}}\mathrm {d} t=\int {\frac {\mathrm {d} t}{t}}=\ln |t|+C=\ln \left|x+{\sqrt {x^{2}+1}}\right|+C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2e2f917892f27619f39a1632f15c0a49d9d30ce4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.988ex; height:8.176ex;" alt="{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {x^{2}+1}}}=\int {\frac {{\frac {1}{2}}+{\frac {1}{2t^{2}}}}{{\frac {t}{2}}+{\frac {1}{2t}}}}\mathrm {d} t=\int {\frac {\mathrm {d} t}{t}}=\ln |t|+C=\ln \left|x+{\sqrt {x^{2}+1}}\right|+C}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Partielle_Integration" title="Partielle Integration">Partielle Integration</a> für eine weitere wichtige Methode zur Berechnung von Integralen,</li>

<li><a href="Weierstra%C3%9F-Substitution" title="Weierstraß-Substitution">Weierstraß-Substitution</a> für bestimmte Funktionen, die <a href="Trigonometrische_Funktionen" class="mw-redirect" title="Trigonometrische Funktionen">trigonometrische Funktionen</a> enthalten.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Harro Heuser: Lehrbuch der Analysis. Teil 1, 5. Auflage, B. G. Teubner, Stuttgart 1988, ISBN 3-519-42221-2, S. 464</li>

<li>Konrad Königsberger: Analysis 1, Springer, Berlin 1992, ISBN 3-540-55116-6, S. 200–201</li>

<li>Richard Courant: Vorlesungen über Differential- und Integralrechnung 1, 4. Auflage, Springer, Berlin / Heidelberg / New York 1971, ISBN 3-540-05466-9, S. 182–191</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://brinkmann-du.de/mathe/gost/diff_int_01_04.htm">Einfache Erklärung/Beispiele für die Substitutionsregel</a></li>

<li>Video: <a rel="nofollow" class="external text" href="https://av.tib.eu/media/9911">Substitutionsregel</a>. <a href="J%C3%B6rn_Loviscach" title="Jörn Loviscach">Jörn Loviscach</a> 2011, zur Verfügung gestellt von der <a href="Technische_Informationsbibliothek" class="mw-redirect" title="Technische Informationsbibliothek">Technischen Informationsbibliothek</a> (TIB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5446/9911">10.5446/9911</a>.</li>

<li>Video: <a rel="nofollow" class="external text" href="https://av.tib.eu/media/10142">Integration durch Substitution, Fingerübung</a>. <a href="J%C3%B6rn_Loviscach" title="Jörn Loviscach">Jörn Loviscach</a> 2013, zur Verfügung gestellt von der <a href="Technische_Informationsbibliothek" class="mw-redirect" title="Technische Informationsbibliothek">Technischen Informationsbibliothek</a> (TIB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5446/10142">10.5446/10142</a>.</li>

<li>Video: <a rel="nofollow" class="external text" href="https://av.tib.eu/media/10144">drei Wege für Integration durch Substitution</a>. <a href="J%C3%B6rn_Loviscach" title="Jörn Loviscach">Jörn Loviscach</a> 2013, zur Verfügung gestellt von der <a href="Technische_Informationsbibliothek" class="mw-redirect" title="Technische Informationsbibliothek">Technischen Informationsbibliothek</a> (TIB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5446/10144">10.5446/10144</a>.</li>

<li>Video: <a rel="nofollow" class="external text" href="https://av.tib.eu/media/9987">Partielle Integration, Substitutionsregel, Integration durch Partialbruchzerlegung</a>. <a href="J%C3%B6rn_Loviscach" title="Jörn Loviscach">Jörn Loviscach</a> 2012, zur Verfügung gestellt von der <a href="Technische_Informationsbibliothek" class="mw-redirect" title="Technische Informationsbibliothek">Technischen Informationsbibliothek</a> (TIB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5446/9987">10.5446/9987</a>.</li>

<li>Video: <a rel="nofollow" class="external text" href="https://av.tib.eu/media/9988">Beispiele partielle Integration, Substitutionsregel, Integration durch Partialbruchzerlegung</a>. <a href="J%C3%B6rn_Loviscach" title="Jörn Loviscach">Jörn Loviscach</a> 2012, zur Verfügung gestellt von der <a href="Technische_Informationsbibliothek" class="mw-redirect" title="Technische Informationsbibliothek">Technischen Informationsbibliothek</a> (TIB), <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5446/9988">10.5446/9988</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> <a href="Otto_Forster" title="Otto Forster">Otto Forster</a>, Florian Lindemann: <cite style="font-style:italic">Analysis 1</cite>. 13. Auflage. Springer Spektrum, 2023, ISBN 978-3-658-40129-0, S. 307.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Integration+durch+Substitution&rft.au=Otto+Forster%2C+Florian+Lindemann&rft.btitle=Analysis+1&rft.date=2023&rft.edition=13.&rft.genre=book&rft.isbn=9783658401290&rft.pages=307&rft.pub=Springer+Spektrum" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a href="Richard_Courant" title="Richard Courant">Richard Courant</a>: <cite style="font-style:italic">Vorlesungen über Differential- und Integralrechnung 1</cite>. 4. Auflage. Springer, 1971, ISBN 3-540-05466-9, S. 184.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Integration+durch+Substitution&rft.au=Richard+Courant&rft.btitle=Vorlesungen+%C3%BCber+Differential-+und+Integralrechnung+1&rft.date=1971&rft.edition=4.&rft.genre=book&rft.isbn=3540054669&rft.pages=184&rft.pub=Springer" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> <a href="Theodor_Br%C3%B6cker" title="Theodor Bröcker">Theodor Bröcker</a>: <cite style="font-style:italic">Analysis. 1</cite>. 2. Auflage. Spektrum, Akad. Verl, Heidelberg Berlin 1999, ISBN 3-86025-417-0, S. 102.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Integration+durch+Substitution&rft.au=Theodor+Br%C3%B6cker&rft.btitle=Analysis.+1&rft.date=1999&rft.edition=2.&rft.genre=book&rft.isbn=3860254170&rft.pages=102&rft.place=Heidelberg+Berlin&rft.pub=Spektrum%2C+Akad.+Verl" style="display:none">

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-10-17" href="https://de.wikipedia.org/wiki/?title=Integration_durch_Substitution&oldid=260685585">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

</div>

</div></div>

</div>

</main>

</div>

</body></html>