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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Weierstraß-Substitution</span></h1>
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<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"><p>Die <b>Weierstraß-Substitution</b> (auch unter Halbwinkelmethode bekannt) ist eine Methode aus dem <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebiet der <a href="Analysis" title="Analysis">Analysis</a>. Sie ist eine Variante der <a href="Integration_durch_Substitution" title="Integration durch Substitution">Integration durch Substitution</a>, die auf bestimmte Integranden mit <a href="Trigonometrische_Funktion" title="Trigonometrische Funktion">trigonometrischen Funktionen</a> angewendet werden kann. Benannt ist die Methode nach dem Mathematiker <a href="Karl_Weierstra%C3%9F" title="Karl Weierstraß">Karl Weierstraß</a>, der sie entwickelte.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>

<div class="mw-heading mw-heading2"><h2 id="Beschreibung_der_Substitution">Beschreibung der Substitution</h2></div>
<p>Seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>&lt;</mo>
<mi>b</mi>
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<annotation encoding="application/x-tex">{\displaystyle a&lt;b}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91a7698e4c7401bb321f97888b872b583a9e4642.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a<b}" loading="lazy"></span> zwei reelle Zahlen und <span class="mwe-math-element mwe-math-element-inline"><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>
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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> eine <a href="Rationale_Funktion" title="Rationale Funktion">rationale Funktion</a>. Um ein Integral der Form
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{a}^{b}R(\sin(x),\cos(x))\,dx}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
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</msubsup>
<mi>R</mi>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
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<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{b}R(\sin(x),\cos(x))\,dx}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f1883f3cafe20cfd810178de17258b220ddb05c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.573ex; height:6.343ex;" alt="{\displaystyle \int _{a}^{b}R(\sin(x),\cos(x))\,dx}" loading="lazy"></span></dd></dl>
<p>zu berechnen, kann die Substitution
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan \left({\frac {x}{2}}\right)=t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tan</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tan \left({\frac {x}{2}}\right)=t}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a56bd47b2b92cb4dc6a4206f12bee1774c59738.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.239ex; height:4.843ex;" alt="{\displaystyle \tan \left({\frac {x}{2}}\right)=t}" loading="lazy"></span></dd></dl>
<p>für <span class="mwe-math-element mwe-math-element-inline"><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|<\pi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
</mrow>
<mo>&lt;</mo>
<mi>π<!-- π --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle |x|&lt;\pi }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/97cbf9c7ee42215021450795e256e3020d020b47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.054ex; height:2.843ex;" alt="{\displaystyle |x|<\pi }" loading="lazy"></span> angewandt werden. Für die Funktionen <a href="Sinus" class="mw-redirect" title="Sinus">Sinus</a> und <a href="Kosinus" class="mw-redirect" title="Kosinus">Kosinus</a> ergeben sich dann die Substitutionen
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}\sin x&amp;={\frac {2t}{1+t^{2}}}\\\cos x&amp;={\frac {1-t^{2}}{1+t^{2}}}\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>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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<mtr>
<mtd>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mi>x</mi>
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<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
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<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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</mfrac>
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</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sin x&amp;={\frac {2t}{1+t^{2}}}\\\cos x&amp;={\frac {1-t^{2}}{1+t^{2}}}\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f89e865aee36dd9748a9500863e02629d0a1d522.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:15.411ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\sin x&amp;={\frac {2t}{1+t^{2}}}\\\cos x&amp;={\frac {1-t^{2}}{1+t^{2}}}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>und für das <a href="Differential_(Mathematik)" title="Differential (Mathematik)">Differential</a> gilt
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 {2\,dt}{1+t^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>x</mi>
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<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
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<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle dx={\frac {2\,dt}{1+t^{2}}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07dfc92fe00e8f901ec37aa875218ec2008aa57b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.377ex; height:5.843ex;" alt="{\displaystyle dx={\frac {2\,dt}{1+t^{2}}}}" loading="lazy"></span>.</dd></dl>
<p>Da sich die Funktionen <a href="Tangens" class="mw-redirect" title="Tangens">Tangens</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tan }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>tan</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \tan }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b57c91704d9ab0366a6436869e2968491efc5155.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.36ex; height:2.009ex;" alt="{\displaystyle \tan }" loading="lazy"></span>, <a href="Kotangens" class="mw-redirect" title="Kotangens">Kotangens</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cot }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cot</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cot }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5bce4c1822b49c4e7ecd7b9aa07f2fcb1706817.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.099ex; height:2.009ex;" alt="{\displaystyle \cot }" loading="lazy"></span>, <a href="Sekans" class="mw-redirect" title="Sekans">Sekans</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sec }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sec</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sec }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dd93d9784c55e8ef40f81993114bd3c3d01084c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \sec }" loading="lazy"></span> und <a href="Kosekans" class="mw-redirect" title="Kosekans">Kosekans</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \csc }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>csc</mi>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle \csc }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/399542cf3e9a873371132ffebe579187d48962ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.981ex; height:1.676ex;" alt="{\displaystyle \csc }" loading="lazy"></span> als Brüche mit Sinus und Kosinus schreiben lassen, kann auch auf diese trigonometrischen Funktionen die Weierstraß-Substitution angewandt werden. Die Substitutionen lauten
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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}\tan x&amp;={\frac {2t}{1-t^{2}}}\\\cot x&amp;={\frac {1-t^{2}}{2t}}\\\csc x&amp;={\frac {1+t^{2}}{2t}}\\\sec x&amp;={\frac {1+t^{2}}{1-t^{2}}}\,.\end{aligned}}}">
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<mtr>
<mtd>
<mi>csc</mi>
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<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
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<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\tan x&amp;={\frac {2t}{1-t^{2}}}\\\cot x&amp;={\frac {1-t^{2}}{2t}}\\\csc x&amp;={\frac {1+t^{2}}{2t}}\\\sec x&amp;={\frac {1+t^{2}}{1-t^{2}}}\,.\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e3cd859ea90562c23dc7bb32314d328002137efd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.171ex; width:16.693ex; height:23.509ex;" alt="{\displaystyle {\begin{aligned}\tan x&amp;={\frac {2t}{1-t^{2}}}\\\cot x&amp;={\frac {1-t^{2}}{2t}}\\\csc x&amp;={\frac {1+t^{2}}{2t}}\\\sec x&amp;={\frac {1+t^{2}}{1-t^{2}}}\,.\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Alternativ kann ein Integral von der obigen Form auch auf <a href="Funktionentheorie" title="Funktionentheorie">funktionentheoretische</a> Weise gelöst werden. Dabei wird das reelle Intervall in ein komplexes Gebiet transformiert und anschließend der <a href="Residuensatz#Trigonometrische_Funktionen" title="Residuensatz"> Residuensatz</a> angewendet.
</p>
<div class="mw-heading mw-heading2"><h2 id="Beispiel">Beispiel</h2></div>
<p>Die Generalsubstitution ist geeignet, die trigonometrischen Funktionen bei der Berechnung des Integrals zu eliminieren, wie das folgende Beispiel zeigt.
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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 {2}{3+\cos(x)}}\,dx=\int {\frac {2}{(3+{\frac {1-t^{2}}{1+t^{2}}})}}\cdot {\frac {2}{(1+t^{2})}}\,dt=\int {\frac {4}{3+3t^{2}+1-t^{2}}}\,dt=\int {\frac {2}{2+t^{2}}}\,dt}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mrow>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo>⋅<!-- ⋅ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>4</mn>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mn>3</mn>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>+</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mrow>
<mn>2</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int {\frac {2}{3+\cos(x)}}\,dx=\int {\frac {2}{(3+{\frac {1-t^{2}}{1+t^{2}}})}}\cdot {\frac {2}{(1+t^{2})}}\,dt=\int {\frac {4}{3+3t^{2}+1-t^{2}}}\,dt=\int {\frac {2}{2+t^{2}}}\,dt}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e11b1b3d296a1a5373d39f1da2908bee7ca55a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:85.565ex; height:7.676ex;" alt="{\displaystyle \int {\frac {2}{3+\cos(x)}}\,dx=\int {\frac {2}{(3+{\frac {1-t^{2}}{1+t^{2}}})}}\cdot {\frac {2}{(1+t^{2})}}\,dt=\int {\frac {4}{3+3t^{2}+1-t^{2}}}\,dt=\int {\frac {2}{2+t^{2}}}\,dt}" loading="lazy"></span></dd></dl>
<p>Dieses Integral lässt sich nun mit einer weiteren <a href="Integration_durch_Substitution" title="Integration durch Substitution">Integration durch Substitution</a> berechnen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2></div>
<p>In diesem Abschnitt werden die Substitutionsformeln für Sinus und Kosinus hergeleitet. Mit den <a href="Formelsammlung_Trigonometrie#Additionstheoreme" title="Formelsammlung Trigonometrie">Additionstheoremen</a> erhält man:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)=2\sin \left({\frac {x}{2}}\right)\cos \left({\frac {x}{2}}\right)=2t\cdot \cos ^{2}\left({\frac {x}{2}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>t</mi>
<mo>⋅<!-- ⋅ --></mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin(x)=2\sin \left({\frac {x}{2}}\right)\cos \left({\frac {x}{2}}\right)=2t\cdot \cos ^{2}\left({\frac {x}{2}}\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d3bd6894ca04389b5b99789237d1ea65709b1a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.765ex; height:4.843ex;" alt="{\displaystyle \sin(x)=2\sin \left({\frac {x}{2}}\right)\cos \left({\frac {x}{2}}\right)=2t\cdot \cos ^{2}\left({\frac {x}{2}}\right)}" loading="lazy"></span></dd></dl>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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={\frac {\sin({\frac {x}{2}})}{\cos({\frac {x}{2}})}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos ^{2}\left({\frac {x}{2}}\right)}{t^{2}}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1}{1+t^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mfrac>
</mrow>
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<msup>
<mi>cos</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t={\frac {\sin({\frac {x}{2}})}{\cos({\frac {x}{2}})}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos ^{2}\left({\frac {x}{2}}\right)}{t^{2}}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1}{1+t^{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ecd5485743f8c889c879a35ec1378b27d049bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:65.716ex; height:7.343ex;" alt="{\displaystyle t={\frac {\sin({\frac {x}{2}})}{\cos({\frac {x}{2}})}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1-\cos ^{2}\left({\frac {x}{2}}\right)}{t^{2}}}\Longrightarrow \cos ^{2}\left({\frac {x}{2}}\right)={\frac {1}{1+t^{2}}}}" loading="lazy"></span>.</dd></dl>
<p>Zusammen hat man die Darstellung oben für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a990a5545cac26c1c6821dca95d898bc80fe3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}" loading="lazy"></span>.
Die Darstellung für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eb9af7ed6f44822021b74bb82b431022c7fd66b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.25ex; height:2.843ex;" alt="{\displaystyle \cos(x)}" loading="lazy"></span> erhält man wie folgt:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)={\sqrt {1-\sin ^{2}(x)}}={\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos(x)={\sqrt {1-\sin ^{2}(x)}}={\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/645bcf6417da827c140f79b1c1c27c5a7600d480.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:51.577ex; height:7.676ex;" alt="{\displaystyle \cos(x)={\sqrt {1-\sin ^{2}(x)}}={\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1456d52196a1482372d7760aa863f111d53aa824.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.544ex; height:4.843ex;" alt="{\displaystyle x\in \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]}" loading="lazy"></span>,</dd>
<dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos(x)=-{\sqrt {1-\sin ^{2}(x)}}=-{\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>cos</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>sin</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<msqrt>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>4</mn>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</msqrt>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \cos(x)=-{\sqrt {1-\sin ^{2}(x)}}=-{\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/88af57e4a333f6390eaad6cf923876a40dee182b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:55.193ex; height:7.676ex;" alt="{\displaystyle \cos(x)=-{\sqrt {1-\sin ^{2}(x)}}=-{\sqrt {1-{\frac {4t^{2}}{(1+t^{2})^{2}}}}}={\frac {1-t^{2}}{1+t^{2}}}}" loading="lazy"></span> für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>[</mo>
<mrow>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>3</mn>
<mi>π<!-- π --></mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b53a896dfa7e6cb30b789625e63eec3bd53faa6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.158ex; height:6.176ex;" alt="{\displaystyle x\in \left[{\frac {\pi }{2}},{\frac {3\pi }{2}}\right]}" loading="lazy"></span>.</dd></dl>
<p>Die Ableitung von <span class="mwe-math-element mwe-math-element-inline"><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="./_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"></span> nach <span class="mwe-math-element mwe-math-element-inline"><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="./_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"></span> ergibt sich mit:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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=\tan \left({\frac {x}{2}}\right)\iff x=2\cdot \arctan(t)\Rightarrow {\frac {dx}{dt}}={\frac {2}{1+t^{2}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>=</mo>
<mi>tan</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thickmathspace"></mspace>
<mo stretchy="false">⟺<!-- ⟺ --></mo>
<mspace width="thickmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mo>⋅<!-- ⋅ --></mo>
<mi>arctan</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">⇒<!-- ⇒ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>2</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t=\tan \left({\frac {x}{2}}\right)\iff x=2\cdot \arctan(t)\Rightarrow {\frac {dx}{dt}}={\frac {2}{1+t^{2}}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b816e485a1fdee3ea64c606c2faa87803fe5923.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:52.348ex; height:5.843ex;" alt="{\displaystyle t=\tan \left({\frac {x}{2}}\right)\iff x=2\cdot \arctan(t)\Rightarrow {\frac {dx}{dt}}={\frac {2}{1+t^{2}}}}" loading="lazy"></span>.</dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a href="Eric_Weisstein" title="Eric Weisstein">Eric W. Weisstein</a>: <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/WeierstrassSubstitution.html"><i>Weierstrass Substitution</i>.</a> In: <i><a href="MathWorld" title="MathWorld">MathWorld</a></i> (englisch).</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Howard Anton, Irl Bivens, Stephen Davis: <cite style="font-style:italic">Calculus</cite>. 9. Auflage. John Wiley &amp; Sons, Inc., 2009, ISBN 978-0-470-18345-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>526–528</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Weierstra%C3%9F-Substitution&amp;rft.au=Howard+Anton%2C+Irl+Bivens%2C+Stephen+Davis&amp;rft.btitle=Calculus&amp;rft.date=2009&amp;rft.edition=9.&amp;rft.genre=book&amp;rft.isbn=9780470183458&amp;rft.pages=526-528&amp;rft.pub=John+Wiley+%26+Sons%2C+Inc." style="display:none">&nbsp;</span> </span>
</li>
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