Newton-Cotes-Formeln

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Diesen Formeln liegt die Idee zu Grunde, die zu integrierende <a href="/wiki/Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktion</a> durch ein <a href="/wiki/Polynom" title="Polynom">Polynom</a> zu <a href="/wiki/Polynominterpolation" title="Polynominterpolation">interpolieren</a> und dieses als Näherung exakt zu integrieren. Die <a href="/wiki/St%C3%BCtzstelle" title="Stützstelle">Stützstellen</a> der <a href="/wiki/Interpolation_(Mathematik)" title="Interpolation (Mathematik)">Interpolation</a> werden dabei <a href="/wiki/%C3%84quidistanz_(Geometrie)" title="Äquidistanz (Geometrie)">äquidistant</a> gewählt. <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><label class="toctogglelabel" for="toctogglecheckbox"></label></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Herleitung">1 Herleitung</a></li> <li class="toclevel-1 tocsection-2"><a href="#Definition">2 Definition</a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Abgeschlossene_Newton-Cotes-Formeln">2.1 Abgeschlossene Newton-Cotes-Formeln</a></li> <li class="toclevel-2 tocsection-4"><a href="#Offene_Newton-Cotes-Formeln">2.2 Offene Newton-Cotes-Formeln</a></li> <li class="toclevel-2 tocsection-5"><a href="#Maclaurin-Formeln">2.3 Maclaurin-Formeln</a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Summierte_Newton-Cotes-Formeln">3 Summierte Newton-Cotes-Formeln</a></li> <li class="toclevel-1 tocsection-7"><a href="#Literatur">4 Literatur</a></li> <li class="toclevel-1 tocsection-8"><a href="#Einzelnachweise">5 Einzelnachweise</a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Herleitung">Herleitung</h2>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=1" title="Abschnitt bearbeiten: Herleitung" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Herleitung">Quelltext bearbeiten</a>]</div> Für das zu integrierende <a href="/wiki/Interpolationspolynom" class="mw-redirect" title="Interpolationspolynom">Interpolationspolynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a47cca302b556e703b68e2a12168117208f47ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}"> vom Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> werden die Stützstellen <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\leq x_{0}<x_{1}<\dots <x_{n}\leq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≤</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≤</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\leq x_{0}<x_{1}<\dots <x_{n}\leq b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f908ab6a1a065209b94db1214334d34e95d398f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.759ex; height:2.509ex;" alt="{\displaystyle a\leq x_{0}<x_{1}<\dots <x_{n}\leq b}"></dd></dl> äquidistant mit dem konstanten Abstand <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=x_{i+1}-x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h=x_{i+1}-x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bcbeb15d1bffb9c684d57b6e44ab13758a8e44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.637ex; height:2.509ex;" alt="{\displaystyle h=x_{i+1}-x_{i}}"> so gewählt, dass sie symmetrisch zur Intervallmitte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {a+b}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {a+b}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc7810f36b1298823a923ee35d6cd0abe79d122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.69ex; height:3.676ex;" alt="{\displaystyle {\tfrac {a+b}{2}}}"> des Integrationsintervalls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> liegen. Somit gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n-i}=a+b-x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n-i}=a+b-x_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93377694217d1b581b88b05e89cbd335fb694e60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.53ex; height:2.509ex;" alt="{\displaystyle x_{n-i}=a+b-x_{i}}">. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90490911c631400e3be106821aa5bcbc1b1b8133" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.712ex; height:2.009ex;" alt="{\displaystyle x_{0}=a}"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5205b5c212d0229bbbb4e87af574fa2534c9fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.644ex; height:2.509ex;" alt="{\displaystyle x_{n}=b}">) erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervalle</a> der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\tfrac {b-a}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcfd2767a84c86fff60f1d0e8837839a10b32bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.127ex; height:3.509ex;" alt="{\displaystyle h={\tfrac {b-a}{n}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+i\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+i\cdot h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40d83d4809309e020854a89c31325171ff72d50a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.119ex; height:2.509ex;" alt="{\displaystyle x_{i}=a+i\cdot h}">. Diese Formeln werden abgeschlossene Newton-Cotes-Formeln genannt. Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\neq a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}\neq a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42f45a37ac624950455b24d51909c8c9629669a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.712ex; height:2.676ex;" alt="{\displaystyle x_{0}\neq a}"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\neq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\neq b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a9e8b71b8a0879cf153515d69ef190f1248c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.644ex; height:2.676ex;" alt="{\displaystyle x_{n}\neq b}">) erhält man offene Newton-Cotes-Formeln: <ul><li>Wählt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a+h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc406addfb4fac0180b238cb7be9ee7cd41303c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.892ex; height:2.509ex;" alt="{\displaystyle x_{0}=a+h}"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b-h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=b-h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/228fb970553f3dc0e35aef0e261e3312b91d6a61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.824ex; height:2.509ex;" alt="{\displaystyle x_{n}=b-h}">), erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaf4207e44c00a9312fe3e8710efbefcd6eafdc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+2}"> Intervalle der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\tfrac {b-a}{n+2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n+2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e06de0014607f77b19375867a149212ba29cdba6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.36ex; height:3.843ex;" alt="{\displaystyle h={\tfrac {b-a}{n+2}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+(1+i)\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+(1+i)\cdot h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/136bb30fc06ec98887a850a6ab4236b2efb960cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.931ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+(1+i)\cdot h}">. Diese Formeln werden offene Newton-Cotes-Formeln genannt.</li> <li>Wählt man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+{\tfrac {h}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a+{\tfrac {h}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad3eafa29e81e8a612214cbe08a58c5cea8bc78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.336ex; height:3.676ex;" alt="{\displaystyle x_{0}=a+{\tfrac {h}{2}}}"> (und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}=b-{\tfrac {h}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}=b-{\tfrac {h}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eccf6c0f98a56959c9a561fc49c75aaf5e95a7f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.267ex; height:3.676ex;" alt="{\displaystyle x_{n}=b-{\tfrac {h}{2}}}">), erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> Intervalle der Länge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\tfrac {b-a}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\tfrac {b-a}{n+1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4a4ca2ea340141813f79bbf46b3b12ea855186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:8.36ex; height:3.843ex;" alt="{\displaystyle h={\tfrac {b-a}{n+1}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>i</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4499426353c5c2607b27eecd6702d64491fbd1d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.747ex; height:3.509ex;" alt="{\displaystyle x_{i}=a+\left({\tfrac {1}{2}}+i\right)\cdot h}">. Diese Formeln werden <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Maclaurin</a>-Formeln genannt.</li></ul> Zur <a href="/wiki/Numerische_Integration" title="Numerische Integration">numerischen Integration</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}f(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0aa70d1cb4a93ca369ce78aebef928db166fb62a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:9.931ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx}"> wird das Interpolationspolynom <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a47cca302b556e703b68e2a12168117208f47ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}"> der Funktion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> zu den gegebenen Stützstellen herangezogen. Für dieses gilt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1630b44c0b28a062f13a27aa8ae0c022336be4de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; margin-left: -0.089ex; width:22.306ex; height:6.843ex;" alt="{\displaystyle p_{n}(x)=\sum _{i=0}^{n}f(x_{i})l_{i}(x)}">,</dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48ba35055d00bcaa522bca9247c8857730998759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.493ex; height:2.509ex;" alt="{\displaystyle l_{i}}"> die <a href="/wiki/Lagrange-Polynom#Lagrange-Basis" class="mw-redirect" title="Lagrange-Polynom">Lagrange-Basispolynome</a> sind. Daraus folgt: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}f(x_{i}){\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}f(x_{i}){\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9d7eb7f06240bd6cd25df45f2bb272019567a9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:46.799ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}f(x_{i}){\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}">.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=2" title="Abschnitt bearbeiten: Definition" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Definition">Quelltext bearbeiten</a>]</div> Für die Newton-Cotes-Formel folgt dann: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}f(x)\,dx\approx \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>≈</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx\approx \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c199bd043bacbd324a70678019044cbedb2edc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:45.856ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx\approx \int \limits _{a}^{b}p_{n}(x)\,dx=(b-a)\sum _{i=0}^{n}w_{i}f(x_{i})}"></dd></dl> mit den Gewichten <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> </mfrac> </mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca37815b42fca9a9c041c1ce2052b875bbb95b02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:21.999ex; height:9.176ex;" alt="{\displaystyle w_{i}={\frac {1}{b-a}}\int \limits _{a}^{b}l_{i}(x)\,dx}"></dd></dl> Die Gewichte sind symmetrisch, das heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{n-i}=w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{n-i}=w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a556319694114abf80a8f2d18c5eee7f20e61f35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.291ex; height:2.009ex;" alt="{\displaystyle w_{n-i}=w_{i}}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l_{i}(x)=\prod _{\begin{smallmatrix}0\leq j\leq n\\j\neq i\end{smallmatrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}={\frac {(x-x_{0})\cdots (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\cdots (x_{i}-x_{i-1})(x_{i}-x_{i+1})\cdots (x_{i}-x_{n})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mtd> </mtr> </mtable> </mstyle> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l_{i}(x)=\prod _{\begin{smallmatrix}0\leq j\leq n\\j\neq i\end{smallmatrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}={\frac {(x-x_{0})\cdots (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\cdots (x_{i}-x_{i-1})(x_{i}-x_{i+1})\cdots (x_{i}-x_{n})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859b802f531f730373ae543e03200bc0aa67c684" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:73.395ex; height:9.176ex;" alt="{\displaystyle l_{i}(x)=\prod _{\begin{smallmatrix}0\leq j\leq n\\j\neq i\end{smallmatrix}}{\frac {x-x_{j}}{x_{i}-x_{j}}}={\frac {(x-x_{0})\cdots (x-x_{i-1})(x-x_{i+1})\cdots (x-x_{n})}{(x_{i}-x_{0})\cdots (x_{i}-x_{i-1})(x_{i}-x_{i+1})\cdots (x_{i}-x_{n})}}}"></dd></dl> Wegen der speziellen Wahl der Stützstellen integrieren diese Formeln bei ungeradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> Polynome bis zum Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, bei geradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> sogar bis zum Grad <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}"> exakt. Somit sind Newton-Cotes-Formeln mit geradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> (also einer ungeraden Anzahl an Stützstellen) denen mit ungeradem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> vorzuziehen. Diese Eigenschaft nennt man auch Genauigkeitsgrad. Speziell gilt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea78f54e69b72f398cf6077e61c50a05b532d4c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}">, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mn>1</mn> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c661e85b466e9a789817893f003d9a6c20504e74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:63.85ex; height:9.176ex;" alt="{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\int \limits _{a}^{b}1\,dx=b-a=(b-a)\sum _{i=0}^{n}w_{i}\cdot 1=(b-a)\sum _{i=0}^{n}w_{i}}"> und somit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}w_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}w_{i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e14d38c1026d6848808d5f2aab78ba2d10dedfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.467ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}w_{i}=1}">. Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c293a69dedaf58c3a86aa2c7f4dcbf510fbfcfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.065ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|>\sum _{i=0}^{n}w_{i}=1}">, was bei Gewichten mit verschiedenen <a href="/wiki/Vorzeichen_(Zahl)" title="Vorzeichen (Zahl)">Vorzeichen</a> der Fall ist, besteht die Gefahr, dass sich die <a href="/wiki/Rundungsfehler" title="Rundungsfehler">Rundungsfehler</a> aufschaukeln oder <a href="/wiki/Ausl%C3%B6schung_(numerische_Mathematik)" title="Auslöschung (numerische Mathematik)">Auslöschung</a> eintritt. Daher sind aus numerischen Gründen Formeln mit positiven Gewichten zu bevorzugen. Da für großes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> das <a href="/wiki/Interpolationspolynom" class="mw-redirect" title="Interpolationspolynom">Interpolationspolynom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{n}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a47cca302b556e703b68e2a12168117208f47ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.616ex; height:2.843ex;" alt="{\displaystyle p_{n}(x)}"> unbrauchbar ist, sind ebenso Formeln mit großem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> nicht empfehlenswert. Will man bessere Näherungen erreichen, so empfiehlt sich die Verwendung von summierten Formeln. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530d1b0f18a814d822065106b14a0cbac3b725f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:31.774ex; height:9.176ex;" alt="{\displaystyle E(f)=\int \limits _{a}^{b}f(x)\,dx-\int \limits _{a}^{b}p_{n}(x)\,dx}"></dd></dl> ist der Fehler (Verfahrensfehler), der bei der Anwendung der Newton-Cotes-Formel gemacht wird. Dieser hat bei der speziellen Wahl der Stützstellen für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da0746babaf86fc30da6b03fa83177f699790ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle (p+1)}">-mal auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> <a href="/wiki/Stetig_differenzierbar" class="mw-redirect" title="Stetig differenzierbar">stetig differenzierbar</a> reellwertige Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> immer die Form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>K</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a56817275231d4e58ea376c6158009fbe3c2665e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.306ex; height:3.343ex;" alt="{\displaystyle E(f)=K\cdot f^{(p+1)}(\xi )}">,</dd></dl> wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"> eine von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"> unabhängige Konstante und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc05d1fb45b90e25c99bc6a57473d508d3e9c23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.426ex; height:2.843ex;" alt="{\displaystyle \xi \in [a,b]}"> ein nur in Ausnahmefällen bekannter Zwischenwert ist. Wäre er generell bekannt, könnte man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}"> und somit auch das <a href="/wiki/Integralrechnung" title="Integralrechnung">Integral</a> exakt ausrechnen, im Widerspruch zu der Tatsache, dass es unendlich viele Integrale gibt, die man nicht exakt berechnen kann. Der Fehler ist Null für alle Funktionen, deren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p+1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7da0746babaf86fc30da6b03fa83177f699790ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.982ex; height:2.843ex;" alt="{\displaystyle (p+1)}">-te Ableitung Null ist, also für alle Polynome vom Grad kleiner oder gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}">. Somit ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> der Genauigkeitsgrad. Der Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5885ec01d3b5670fd5f88847f32da2b3dd62f60c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p+1}"> wird auch als (polynomiale) Ordnung der Newton-Cotes-Formel bezeichnet. Mit Hilfe des Verfahrensfehlers erhält man die Fehlerabschätzung: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>≤</mo> <mi>ξ</mi> <mo>≤</mo> <mi>b</mi> </mrow> </munder> <mrow> <mo>|</mo> <mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>p</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6ecd0c7545932b7467321dbddafa1b70a758df7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:29.435ex; height:4.843ex;" alt="{\displaystyle |E(f)|\leq |K|\cdot \max _{a\leq \xi \leq b}\left|f^{(p+1)}(\xi )\right|}">.</dd></dl> Der exakte Fehler ist immer kleiner oder gleich dieser Fehlerabschätzung, wie auch die unten angegebenen Beispiele zeigen. <div class="mw-heading mw-heading3"><h3 id="Abgeschlossene_Newton-Cotes-Formeln">Abgeschlossene Newton-Cotes-Formeln</h3>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=3" title="Abschnitt bearbeiten: Abgeschlossene Newton-Cotes-Formeln" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Abgeschlossene Newton-Cotes-Formeln">Quelltext bearbeiten</a>]</div> Die angegebenen Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> gelten für das Integrationsintervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>i</mi> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9c6e38c825624bcaa5b36002f37b438ab1093c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.51ex; height:5.176ex;" alt="{\displaystyle t_{0}=0,t_{i}={\frac {i}{n}},t_{n}=1}">. Für ein allgemeines Intervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe87fb95b4dd8d76d1700605f2575d06407f0865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.493ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">. <table class="wikitable"> <tbody><tr> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </th> <th class="hintergrundfarbe6">Name </th> <th class="hintergrundfarbe6">Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> </th> <th class="hintergrundfarbe6">Gewichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"> </th> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}"> </th></tr> <tr> <td>1</td> <td><a href="/wiki/Trapezregel" title="Trapezregel">Trapezregel</a> Sehnentrapezregel </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/252ae6fabd6a414a413dfa7790fa5b0cd163d53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.648ex; height:2.176ex;" alt="{\displaystyle 0\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bca2d9a583274c41fec309ea88cd6258e533adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>12</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98523454e05921fcb510fe16505ff06d78ec6e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.873ex; height:5.843ex;" alt="{\displaystyle -{\frac {(b-a)^{3}}{12}}f''(\xi )}"> </td></tr> <tr> <td>2</td> <td><a href="/wiki/Simpsonregel" title="Simpsonregel">Simpson-Regel</a> <a href="/wiki/Keplersche_Fassregel" class="mw-redirect" title="Keplersche Fassregel">Keplersche Fassregel</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad {\frac {1}{2}}\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{2}}\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bdc5c985e67e9d3c570c141e7bda091c2fa8984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.969ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{2}}\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab87ddb4243faa01c77bff0356422e2d8a5577e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {4}{6}}\quad {\frac {1}{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mn>90</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de11b4f9184cd0d5969bf132e33218cb50c14b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.657ex; height:8.009ex;" alt="{\displaystyle -{\frac {\left({\frac {b-a}{2}}\right)^{5}}{90}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>3</td> <td>3/8-Regel Pulcherrima </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3ad772df42f60fb64a80b734f02104a5eb02ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.29ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{3}}\quad {\frac {2}{3}}\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37c218d021a2e30acf6b27d214a2b66e1402f73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {3}{8}}\quad {\frac {1}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>80</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab897c075754eb1e96e592c5accd9988165be82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.82ex; height:8.009ex;" alt="{\displaystyle -{\frac {3\left({\frac {b-a}{3}}\right)^{5}}{80}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>4</td> <td><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Milne</a>-Regel <a href="/wiki/George_Boole" title="George Boole">Boole</a>-Regel </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58695b3e100f5b2cc57f5812ff625486a2292065" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.611ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>90</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>90</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mn>90</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>32</mn> <mn>90</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>90</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5661da12df91d0afce5c65eaa9db84416170b496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.096ex; height:5.176ex;" alt="{\displaystyle {\frac {7}{90}}\quad {\frac {32}{90}}\quad {\frac {12}{90}}\quad {\frac {32}{90}}\quad {\frac {7}{90}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mn>945</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf4e5083f82d3c6fc8ba516dfc2e11978097bc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.82ex; height:8.176ex;" alt="{\displaystyle -{\frac {8\left({\frac {b-a}{4}}\right)^{7}}{945}}f^{(6)}(\xi )}"> </td></tr> <tr> <td>5</td> <td>6-Punkt-Regel </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb147678b551cc4112939e686630e03409e201cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.932ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mn>288</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>75</mn> <mn>288</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>50</mn> <mn>288</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>50</mn> <mn>288</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>75</mn> <mn>288</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>19</mn> <mn>288</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9c4ea3bc3f85586d88ae31bfb37c81c459bbb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.554ex; height:5.176ex;" alt="{\displaystyle {\frac {19}{288}}\quad {\frac {75}{288}}\quad {\frac {50}{288}}\quad {\frac {50}{288}}\quad {\frac {75}{288}}\quad {\frac {19}{288}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>275</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>12</mn> <mspace width="thinmathspace" /> <mn>096</mn> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb16f3f1acec30dcabe52cb5adee1be14b79cedf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.145ex; height:8.009ex;" alt="{\displaystyle -{\frac {275\left({\frac {b-a}{5}}\right)^{7}}{12\,096}}f^{(6)}(\xi )}"> </td></tr> <tr> <td>6</td> <td>Weddle-Regel (nach <a href="/w/index.php?title=Thomas_Weddle&action=edit&redlink=1" class="new" title="Thomas Weddle (Seite nicht vorhanden)">Thomas Weddle</a>, 1817–1853)<a href="#cite_note-1">[1]</a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb374857687bbabee663985515169b3e96b5f3e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.253ex; height:5.176ex;" alt="{\displaystyle 0\quad {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}\quad 1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>41</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>216</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>27</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>272</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>27</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>216</mn> <mn>840</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>41</mn> <mn>840</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d79f5235ba0f676d61a6e11e5b7e66d3a3428a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.2ex; height:5.343ex;" alt="{\displaystyle {\frac {41}{840}}\quad {\frac {216}{840}}\quad {\frac {27}{840}}\quad {\frac {272}{840}}\quad {\frac {27}{840}}\quad {\frac {216}{840}}\quad {\frac {41}{840}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <mn>1400</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a26ccc184f3263973f4994ef798dff7d4709dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.82ex; height:8.176ex;" alt="{\displaystyle -{\frac {9\left({\frac {b-a}{6}}\right)^{9}}{1400}}f^{(8)}(\xi )}"> </td></tr></tbody></table> Die gekürzten Werte aller Gewichte bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fecadd7726f4d4c6c390c2f2e73533d7a3729ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=10}"> betragen:<a href="#cite_note-2">[2]</a> <table class="wikitable" style="text-align:right"> <tbody><tr> <th>n </th> <th>Gewichte </th></tr> <tr> <th>1 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/046af4af8c4f600a2c4c17d74aeee5a66dfc27a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.675ex; height:3.509ex;" alt="{\displaystyle \{{\tfrac {1}{2}},{\tfrac {1}{2}}\}}"> </td></tr> <tr> <th>2 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a11e2339a75cbacf99e81cf30e1d5916012159" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.367ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {1}{6}},{\tfrac {2}{3}},{\tfrac {1}{6}}\}}"> </td></tr> <tr> <th>3 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd609fa2647cd14195bccb0c753eed723ef444a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.059ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {3}{8}},{\tfrac {1}{8}}\}}"> </td></tr> <tr> <th>4 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>90</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>45</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>15</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16</mn> <mn>45</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>7</mn> <mn>90</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1711be765f4c0947caffc50b572bc6613574924f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:18.861ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {7}{90}},{\tfrac {16}{45}},{\tfrac {2}{15}},{\tfrac {16}{45}},{\tfrac {7}{90}}\}}"> </td></tr> <tr> <th>5 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>19</mn> <mn>288</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>96</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>144</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>144</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>25</mn> <mn>96</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>19</mn> <mn>288</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813f9ad8eda634091b66b9344542d5f531dc560f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:25.663ex; height:3.676ex;" alt="{\displaystyle \{{\tfrac {19}{288}},{\tfrac {25}{96}},{\tfrac {25}{144}},{\tfrac {25}{144}},{\tfrac {25}{96}},{\tfrac {19}{288}}\}}"> </td></tr> <tr> <th>6 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>41</mn> <mn>840</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>35</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>34</mn> <mn>105</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>9</mn> <mn>35</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>41</mn> <mn>840</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d27f72fc20944236ba78d75fbcab157244c3c1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:29.999ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {41}{840}},{\tfrac {9}{35}},{\tfrac {9}{280}},{\tfrac {34}{105}},{\tfrac {9}{280}},{\tfrac {9}{35}},{\tfrac {41}{840}}\}}"> </td></tr> <tr> <th>7 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>751</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3577</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>49</mn> <mn>640</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2989</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2989</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>49</mn> <mn>640</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3577</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>751</mn> <mn>17280</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41b006ac19cad94bb14ab024d3b68bc471732714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:45.843ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {751}{17280}},{\tfrac {3577}{17280}},{\tfrac {49}{640}},{\tfrac {2989}{17280}},{\tfrac {2989}{17280}},{\tfrac {49}{640}},{\tfrac {3577}{17280}},{\tfrac {751}{17280}}\}}"> </td></tr> <tr> <th>8 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>989</mn> <mn>28350</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2944</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>464</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5248</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>454</mn> <mn>2835</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5248</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>464</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2944</mn> <mn>14175</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>989</mn> <mn>28350</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c3b8adac3ff19b7a9a7824124b672ce6addde2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:59.713ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {989}{28350}},{\tfrac {2944}{14175}},-{\tfrac {464}{14175}},{\tfrac {5248}{14175}},-{\tfrac {454}{2835}},{\tfrac {5248}{14175}},-{\tfrac {464}{14175}},{\tfrac {2944}{14175}},{\tfrac {989}{28350}}\}}"> </td></tr> <tr> <th>9 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2857</mn> <mn>89600</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>15741</mn> <mn>89600</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>27</mn> <mn>2240</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1209</mn> <mn>5600</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2889</mn> <mn>44800</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2889</mn> <mn>44800</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1209</mn> <mn>5600</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>27</mn> <mn>2240</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>15741</mn> <mn>89600</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2857</mn> <mn>89600</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c651e4f9e8b851f3e118aa2a32c52dc2f4b7be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:57.803ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {2857}{89600}},{\tfrac {15741}{89600}},{\tfrac {27}{2240}},{\tfrac {1209}{5600}},{\tfrac {2889}{44800}},{\tfrac {2889}{44800}},{\tfrac {1209}{5600}},{\tfrac {27}{2240}},{\tfrac {15741}{89600}},{\tfrac {2857}{89600}}\}}"> </td></tr> <tr> <th>10 </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16067</mn> <mn>598752</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>26575</mn> <mn>149688</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16175</mn> <mn>199584</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5675</mn> <mn>12474</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4825</mn> <mn>11088</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>17807</mn> <mn>24948</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>4825</mn> <mn>11088</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>5675</mn> <mn>12474</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16175</mn> <mn>199584</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>26575</mn> <mn>149688</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>16067</mn> <mn>598752</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0585d7cfd837a9b331124fd946a39f1500412d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:79.235ex; height:3.843ex;" alt="{\displaystyle \{{\tfrac {16067}{598752}},{\tfrac {26575}{149688}},-{\tfrac {16175}{199584}},{\tfrac {5675}{12474}},-{\tfrac {4825}{11088}},{\tfrac {17807}{24948}},-{\tfrac {4825}{11088}},{\tfrac {5675}{12474}},-{\tfrac {16175}{199584}},{\tfrac {26575}{149688}},{\tfrac {16067}{598752}}\}}"> </td></tr></tbody></table> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1344e2c194f016e12bb978493013955fe2f445f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}<0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00eb944de69d6e005c78a17633d5b22d4616ba2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.725ex; height:2.509ex;" alt="{\displaystyle w_{i}<0}"> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=2,4,6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=2,4,6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6f81bff9d8706e5da992bb706f8047f4145847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle i=2,4,6}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>45</mn> <mo>…</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c3b3e778610a00c5d311459f5bc7389b3b7656" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.842ex; height:3.176ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|=1{,}45\dots }"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fecadd7726f4d4c6c390c2f2e73533d7a3729ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.818ex; height:2.176ex;" alt="{\displaystyle n=10}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>3,064</mn> <mn>794</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5d56ba953d519c1f66a30c59a8b97ef77c4b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.492ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}064794\dots }"> Beispiel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6359d53d9a19a61d2c209f4cdda4c6c2c31eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.443ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> Näherung mit Simpson-Regel (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}">). Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ef768ef226e1b051fb7ce3dc7d8e7c6e1dda1f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.699ex; height:5.343ex;" alt="{\displaystyle h={\frac {b-a}{n}}={\frac {2}{2}}=1}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab35f29660dae82321d61edd6aa69a2f4b656a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.973ex; height:2.509ex;" alt="{\displaystyle x_{0}=a=1}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84e953ca2ae4944c3bb0ee862a62587d401b11a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:88.344ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {1}{6}}f(1)+{\frac {4}{6}}f(2)+{\frac {1}{6}}f(3)\right)=2\cdot \left({\frac {1}{6}}\cdot 1+{\frac {4}{6}}\cdot {\frac {1}{2}}+{\frac {1}{6}}\cdot {\frac {1}{3}}\right)={\frac {10}{9}}=1{,}{\overline {1}}}"></dd></dl> Verfahrensfehler: Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>90</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>15</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced64c025ebc2a0beb0e12d6bdf04d7c1f8d80a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.358ex; height:6.509ex;" alt="{\displaystyle E(f)=-{\frac {1}{90}}\cdot \left({\frac {2}{2}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}=-{\frac {4}{15}}\cdot {\frac {1}{\xi ^{5}}}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b658b27a1b1babd01aa16dc7426ceddb92d85fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle \xi \in [1,3]}"> Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>15</mn> </mfrac> </mrow> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>ξ</mi> <mo>≤</mo> <mn>3</mn> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>15</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd242cc5d2742ce72415df0eb536281a55477744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.915ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {4}{15}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {4}{15}}\cdot {\frac {1}{1}}=0{,}2{\overline {6}}}"> Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo>˘</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0,012</mn> <mn>498</mn> <mo>…</mo> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>6</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db921b35725ec99bbb8df503809830c8473bd328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.729ex; height:9.176ex;" alt="{\displaystyle {\breve {a}}|E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}{\overline {1}}\right|=0{,}012498\ldots <0{,}2{\overline {6}}}"> <div class="mw-heading mw-heading3"><h3 id="Offene_Newton-Cotes-Formeln">Offene Newton-Cotes-Formeln</h3>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=4" title="Abschnitt bearbeiten: Offene Newton-Cotes-Formeln" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Offene Newton-Cotes-Formeln">Quelltext bearbeiten</a>]</div> Die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> gelten für das Integrationsintervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/273615f07504a185c664b09c4b55fc5fd4af3b4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.723ex; height:3.843ex;" alt="{\displaystyle t_{0}={\tfrac {1}{n+2}},t_{i}={\tfrac {i+1}{n+2}},t_{n}={\tfrac {n+1}{n+2}}}">. Für ein allgemeines <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe87fb95b4dd8d76d1700605f2575d06407f0865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.493ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">. <table class="wikitable"> <tbody><tr> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </th> <th class="hintergrundfarbe6">Name </th> <th class="hintergrundfarbe6">Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> </th> <th class="hintergrundfarbe6">Gewichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"> </th> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}"> </th></tr> <tr> <td>0</td> <td><a href="/wiki/Rechteckregel" class="mw-redirect" title="Rechteckregel">Rechteckregel</a> <a href="/wiki/Mittelpunktsregel" title="Mittelpunktsregel">Mittelpunktsregel</a> <a href="/wiki/Tangententrapezregel" class="mw-redirect" title="Tangententrapezregel">Tangententrapezregel</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524372005505bc6ed175f1672cbb2076db1229c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.485ex; height:2.176ex;" alt="{\displaystyle 1\quad }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>24</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e865534038ac36a35cf1d0823eaf519eaad6fdeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.064ex; height:5.843ex;" alt="{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}"> </td></tr> <tr> <td>1</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/341b8b1c3600240e22e0d2f42e65c45a0a91b5bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}\quad {\frac {2}{3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bca2d9a583274c41fec309ea88cd6258e533adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>4</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca578c0e5ed12947a5f3f743c754738f6e8f12b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.815ex; height:8.009ex;" alt="{\displaystyle {\frac {3\left({\frac {b-a}{3}}\right)^{3}}{4}}f''(\xi )}"> </td></tr> <tr> <td>2</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d107d587b5fb3303734a8f3a8ea6a6be1ac46e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}\quad {\frac {2}{4}}\quad {\frac {3}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4bdd149a4007c48c240fa68d519d580ee2f8e6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.481ex; height:5.176ex;" alt="{\displaystyle {\frac {2}{3}}\quad -{\frac {1}{3}}\quad {\frac {2}{3}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>14</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>45</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d873fa8c05755ade06c06380a2a7f01c296053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {14\left({\frac {b-a}{4}}\right)^{5}}{45}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>3</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>5</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>5</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5935a09c12de72aed9a7f22ed57bac0cd95e3874" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{5}}\quad {\frac {2}{5}}\quad {\frac {3}{5}}\quad {\frac {4}{5}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>24</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>24</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512517d87e128f5e6dc19d2d7577c3360d0a47e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.612ex; height:5.176ex;" alt="{\displaystyle {\frac {11}{24}}\quad {\frac {1}{24}}\quad {\frac {1}{24}}\quad {\frac {11}{24}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>95</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>144</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f99abe85a90d29ce2564f4190c36001e328369e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.174ex; height:8.009ex;" alt="{\displaystyle {\frac {95\left({\frac {b-a}{5}}\right)^{5}}{144}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>4</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/195c6149dbdef44e477683f0a27618f32cb8f881" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.283ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {2}{6}}\quad {\frac {3}{6}}\quad {\frac {4}{6}}\quad {\frac {5}{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>20</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>20</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>26</mn> <mn>20</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>20</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>20</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82706ff7457dd6ed7660449d92e2b66599327dd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.776ex; height:5.176ex;" alt="{\displaystyle {\frac {11}{20}}\quad -{\frac {14}{20}}\quad {\frac {26}{20}}\quad -{\frac {14}{20}}\quad {\frac {11}{20}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>41</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mn>140</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca763bd582023b12a2b0ea712d995e520d32ddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {41\left({\frac {b-a}{6}}\right)^{7}}{140}}f^{(6)}(\xi )}"> </td></tr> <tr> <td>5</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>7</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>7</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1b009b0c5bbfa06f2edf2a6543a20f0a5c4c40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.604ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{7}}\quad {\frac {2}{7}}\quad {\frac {3}{7}}\quad {\frac {4}{7}}\quad {\frac {5}{7}}\quad {\frac {6}{7}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>611</mn> <mn>1440</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>453</mn> <mn>1440</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>562</mn> <mn>1440</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>562</mn> <mn>1440</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>453</mn> <mn>1440</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>611</mn> <mn>1440</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3cd46163387c2b7f4212a24bf5e73630990675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:50.209ex; height:5.343ex;" alt="{\displaystyle {\frac {611}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {562}{1440}}\quad {\frac {562}{1440}}\quad -{\frac {453}{1440}}\quad {\frac {611}{1440}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5257</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>7</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mn>8640</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffceed2d9c1b0cce5498f5cd0aeb9bdad9949624" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.499ex; height:8.176ex;" alt="{\displaystyle {\frac {5257\left({\frac {b-a}{7}}\right)^{7}}{8640}}f^{(6)}(\xi )}"> </td></tr> <tr> <td>6</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d9e9056c355a6237b8ffed2f947e8b99f21020" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.926ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}\quad {\frac {4}{8}}\quad {\frac {5}{8}}\quad {\frac {6}{8}}\quad {\frac {7}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>460</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>954</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2196</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2459</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2196</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>954</mn> <mn>945</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>460</mn> <mn>945</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5495220ab771f3c18ebc529e5b6d1c2578f31405" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:56.209ex; height:5.343ex;" alt="{\displaystyle {\frac {460}{945}}\quad -{\frac {954}{945}}\quad {\frac {2196}{945}}\quad -{\frac {2459}{945}}\quad {\frac {2196}{945}}\quad -{\frac {954}{945}}\quad {\frac {460}{945}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3956</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>8</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> </mrow> <mrow> <mn>14</mn> <mspace width="thinmathspace" /> <mn>175</mn> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0ee6fb99404d0104b283676652a292aac778f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.499ex; height:8.176ex;" alt="{\displaystyle {\frac {3956\left({\frac {b-a}{8}}\right)^{9}}{14\,175}}f^{(8)}(\xi )}"> </td></tr></tbody></table> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdb41e9a10a8fd7179b9170149a8d70949ba5d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=5}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3252</mn> <mn>1440</mn> </mfrac> </mrow> <mo>=</mo> <mn>2,258</mn> <mn>333</mn> <mo>…</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2ca7a0dead6c7565925ad4ac36298e525f98a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.714ex; height:3.676ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {3252}{1440}}=2{,}258333\dots }"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9679</mn> <mn>945</mn> </mfrac> </mrow> <mo>=</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>24</mn> <mo>…</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7345feb1acf404f42a84bc2608abd4d7e71f9b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:28.227ex; height:3.843ex;" alt="{\displaystyle \textstyle \sum _{i=0}^{n}|w_{i}|={\frac {9679}{945}}=10{,}24\dots }"> Von diesen Formeln ist nur die Rechteckregel empfehlenswert. Die Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"> hat bei höherem Aufwand die gleiche Ordnung wie die Rechteckregel, die höheren Formeln haben negative Gewichte. Beispiel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6359d53d9a19a61d2c209f4cdda4c6c2c31eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.443ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> Näherung mit der Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}">. Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2fa180e0e1182b32ecefa492e7d38e5738f30c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.865ex; height:5.509ex;" alt="{\displaystyle h={\frac {b-a}{n+2}}={\frac {2}{4}}={\frac {1}{2}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+h={\frac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a+h={\frac {3}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2773dcc7faf694225f407f81733dc3c8285098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.989ex; height:5.176ex;" alt="{\displaystyle x_{0}=a+h={\frac {3}{2}}}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>5</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>49</mn> <mn>45</mn> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>8</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1297322df161a7f0efd1e91ddb4aabd4915d307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:97.687ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {2}{3}}f\!\left({\frac {3}{2}}\right)-{\frac {1}{3}}f\!\left({\frac {4}{2}}\right)+{\frac {2}{3}}f\!\left({\frac {5}{2}}\right)\right)=2\cdot \left({\frac {2}{3}}\cdot {\frac {2}{3}}-{\frac {1}{3}}\cdot {\frac {2}{4}}+{\frac {2}{3}}\cdot {\frac {2}{5}}\right)={\frac {49}{45}}=1{,}0{\overline {8}}}">.</dd></dl> Verfahrensfehler: Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>45</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>30</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdad15846309d01d75866d2c73f9cd1048de4fec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.742ex; height:6.509ex;" alt="{\displaystyle E(f)={\frac {14}{45}}\cdot \left({\frac {2}{4}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {7}{30}}\cdot {\frac {1}{\xi ^{5}}}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b658b27a1b1babd01aa16dc7426ceddb92d85fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle \xi \in [1,3]}">. Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>30</mn> </mfrac> </mrow> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>ξ</mi> <mo>≤</mo> <mn>3</mn> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>30</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81dc0942e4f30e6673afcf0e681664b585818087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.915ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {7}{30}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {7}{30}}\cdot {\frac {1}{1}}=0{,}2{\overline {3}}}"> Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>8</mn> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>0,009</mn> <mn>723</mn> <mo>…</mo> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdf669e24c61ffd0ea35a6a4aac413aa5ccbac38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:77.662ex; height:9.176ex;" alt="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=\left|1{,}098612\ldots -1{,}0{\overline {8}}\right|=0{,}009723\ldots <0{,}2{\overline {3}}}"> <div class="mw-heading mw-heading3"><h3 id="Maclaurin-Formeln">Maclaurin-Formeln</h3>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=5" title="Abschnitt bearbeiten: Maclaurin-Formeln" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Maclaurin-Formeln">Quelltext bearbeiten</a>]</div> Diese Formeln sind nach <a href="/wiki/Colin_Maclaurin" title="Colin Maclaurin">Colin Maclaurin</a> benannt. Die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> gelten für das Integrationsintervall <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c07733480eb1226e073fdf36a4264603b26892bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:31.189ex; height:3.843ex;" alt="{\displaystyle t_{0}={\tfrac {1}{2n+2}},t_{i}={\tfrac {2i+1}{2n+2}},t_{n}={\tfrac {2n+1}{2n+2}}}">. Für ein allgemeines <a href="/wiki/Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> sind die Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe87fb95b4dd8d76d1700605f2575d06407f0865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.493ex; height:2.843ex;" alt="{\displaystyle x_{i}=a+t_{i}\cdot (b-a)}">. <table class="wikitable"> <tbody><tr> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> </th> <th class="hintergrundfarbe6">Name </th> <th class="hintergrundfarbe6">Stützstellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b61e3d4d909be4a19c9a554a301684232f59e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.639ex; height:2.343ex;" alt="{\displaystyle t_{i}}"> </th> <th class="hintergrundfarbe6">Gewichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe22f0329d3ecb2e1880d44d191aba0e5475db68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.009ex;" alt="{\displaystyle w_{i}}"> </th> <th class="hintergrundfarbe6"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}"> </th></tr> <tr> <td>0</td> <td><a href="/wiki/Rechteckregel" class="mw-redirect" title="Rechteckregel">Rechteckregel</a> <a href="/wiki/Mittelpunktsregel" title="Mittelpunktsregel">Mittelpunktsregel</a> <a href="/wiki/Tangententrapezregel" class="mw-redirect" title="Tangententrapezregel">Tangententrapezregel</a></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11cfb2fdb143693b1daf78fcb5c11a023cb1c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:1.999ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}}"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/524372005505bc6ed175f1672cbb2076db1229c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.485ex; height:2.176ex;" alt="{\displaystyle 1\quad }"></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mn>24</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e865534038ac36a35cf1d0823eaf519eaad6fdeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.064ex; height:5.843ex;" alt="{\displaystyle {\frac {(b-a)^{3}}{24}}f''(\xi )}"> </td></tr> <tr> <td>1</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfcaae772d169ee58093dbf5c0a82e802dbe5532" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4}}\quad {\frac {3}{4}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bca2d9a583274c41fec309ea88cd6258e533adc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:6.32ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{2}}\quad {\frac {1}{2}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>12</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22135bcb12c85492fcc6d960e315d6df29639a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.653ex; height:8.009ex;" alt="{\displaystyle {\frac {\left({\frac {b-a}{2}}\right)^{3}}{12}}f''(\xi )}"> </td></tr> <tr> <td>2</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>6</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8722e832370090ec115b31d34e6992f74d174677" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{6}}\quad {\frac {1}{2}}\quad {\frac {5}{6}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aeabaf129fbcf05d1b115da32c65bdc88184672" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.641ex; height:5.176ex;" alt="{\displaystyle {\frac {3}{8}}\quad {\frac {2}{8}}\quad {\frac {3}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>21</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>640</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c238ca077e46da3c02d67bcf76ff8f3a8e2b416e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.174ex; height:8.176ex;" alt="{\displaystyle {\frac {21\left({\frac {b-a}{3}}\right)^{5}}{640}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>3</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>8</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4e7a56e6e5319fee0a484920d37a86b66711fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.962ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{8}}\quad {\frac {3}{8}}\quad {\frac {5}{8}}\quad {\frac {7}{8}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>13</mn> <mn>48</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>48</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>48</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>13</mn> <mn>48</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aff60968177fb371b94e9f494dcb0ec44e087678" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.612ex; height:5.343ex;" alt="{\displaystyle {\frac {13}{48}}\quad {\frac {11}{48}}\quad {\frac {11}{48}}\quad {\frac {13}{48}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>103</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mn>1440</mn> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1730d42153aefa82b2be89c8e5d7ba3ac84442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.336ex; height:8.176ex;" alt="{\displaystyle {\frac {103\left({\frac {b-a}{4}}\right)^{5}}{1440}}f^{(4)}(\xi )}"> </td></tr> <tr> <td>4</td> <td></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>10</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>10</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76fc33739ed99f14adb9ad08b2c568c52d831106" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.096ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{10}}\quad {\frac {3}{10}}\quad {\frac {5}{10}}\quad {\frac {7}{10}}\quad {\frac {9}{10}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>275</mn> <mn>1152</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>1152</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>402</mn> <mn>1152</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>100</mn> <mn>1152</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>275</mn> <mn>1152</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcef7a491c54c66f025c4ba5bdd20f45de1e46a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.72ex; height:5.176ex;" alt="{\displaystyle {\frac {275}{1152}}\quad {\frac {100}{1152}}\quad {\frac {402}{1152}}\quad {\frac {100}{1152}}\quad {\frac {275}{1152}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5575</mn> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mn>5</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mrow> <mn>193</mn> <mspace width="thinmathspace" /> <mn>536</mn> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219d846b00539703c7ceca175adebf83adf4b601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.499ex; height:8.009ex;" alt="{\displaystyle {\frac {5575\left({\frac {b-a}{5}}\right)^{7}}{193\,536}}f^{(6)}(\xi )}"> </td></tr></tbody></table> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1,363</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b247046987691fc8f081f517371865e45b65d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.005ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=1{,}363\dots }"> Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1344e2c194f016e12bb978493013955fe2f445f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=8}"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>3,433</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de33375d94e43e134f746d05d204483420d2429" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:19.005ex; height:6.843ex;" alt="{\displaystyle \sum _{i=0}^{n}|w_{i}|=3{,}433\dots }"> Beispiel: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6359d53d9a19a61d2c209f4cdda4c6c2c31eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:46.443ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}{\frac {1}{x}}\,dx=\ln(3)-\ln(1)=\ln(3)=1{,}098612\dots }"> Näherung mit der Formel für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a02c8bd752d2cc859747ca1f3a508281bdbc3b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}">. Es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78422fd209b67d980ccc7b77b330dce1059cde8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.768ex; height:5.509ex;" alt="{\displaystyle h={\frac {b-a}{n+1}}={\frac {2}{3}}}"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>h</mi> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f4213e097eecbde21ead6f785de5aae36e87e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.825ex; height:5.343ex;" alt="{\displaystyle x_{0}=a+{\frac {h}{2}}={\frac {4}{3}}}">. <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>6</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mi>f</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>8</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>6</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>105</mn> <mn>96</mn> </mfrac> </mrow> <mo>=</mo> <mn>1,093</mn> <mn>75</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6844de1f9c1e4ad12d2b8230e0107e6eb0d8ef2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:102.222ex; height:9.176ex;" alt="{\displaystyle \int \limits _{1}^{3}p_{2}(x)\,dx=2\cdot \left({\frac {3}{8}}f\!\left({\frac {4}{3}}\right)+{\frac {2}{8}}f\!\left({\frac {6}{3}}\right)+{\frac {3}{8}}f\!\left({\frac {8}{3}}\right)\right)=2\cdot \left({\frac {3}{8}}\cdot {\frac {3}{4}}+{\frac {2}{8}}\cdot {\frac {3}{6}}+{\frac {3}{8}}\cdot {\frac {3}{8}}\right)={\frac {105}{96}}=1{,}09375}"></dd></dl> Verfahrensfehler: Mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a14231d4b6d9881dec0bc80d5e9a9310bd71fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.516ex; height:6.009ex;" alt="{\displaystyle f^{(4)}(\xi )={\frac {4!}{\xi ^{5}}}}"> erhält man <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>21</mn> <mn>640</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mo>!</mo> </mrow> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>135</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44f9df123978f09dc0c74dbdc61721542e7f13c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.067ex; height:6.509ex;" alt="{\displaystyle E(f)={\frac {21}{640}}\cdot \left({\frac {2}{3}}\right)^{5}\cdot {\frac {4!}{\xi ^{5}}}={\frac {14}{135}}\cdot {\frac {1}{\xi ^{5}}}}"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi \in [1,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi \in [1,3]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b658b27a1b1babd01aa16dc7426ceddb92d85fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle \xi \in [1,3]}">. Fehlerabschätzung: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>135</mn> </mfrac> </mrow> <mo>⋅</mo> <munder> <mo movablelimits="true" form="prefix">max</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>≤</mo> <mi>ξ</mi> <mo>≤</mo> <mn>3</mn> </mrow> </munder> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>14</mn> <mn>135</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>037</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08889179d18036b68d01def771b5f03274366c9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.565ex; height:6.176ex;" alt="{\displaystyle |E(f)|\leq {\frac {14}{135}}\cdot \max _{1\leq \xi \leq 3}\left|{\frac {1}{\xi ^{5}}}\right|={\frac {14}{135}}\cdot {\frac {1}{1}}=0{,}1{\overline {037}}}"> Exakter Fehler: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>−</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>1,098</mn> <mn>612</mn> <mo>…</mo> <mo>−</mo> <mn>1,093</mn> <mn>75</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0,000</mn> <mn>486</mn> <mo>…</mo> <mo><</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>,</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>037</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27cba58bfb6e658597b4a67a34af6c2ec70faef5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:83.359ex; height:9.176ex;" alt="{\displaystyle |E(f)|=\left|\int \limits _{1}^{3}{\frac {1}{x}}\,dx-\int \limits _{1}^{3}p_{2}(x)\,dx\right|=|1{,}098612\ldots -1{,}09375|=0{,}000486\ldots <0{,}1{\overline {037}}}"> <div class="mw-heading mw-heading2"><h2 id="Summierte_Newton-Cotes-Formeln">Summierte Newton-Cotes-Formeln</h2>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=6" title="Abschnitt bearbeiten: Summierte Newton-Cotes-Formeln" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Summierte Newton-Cotes-Formeln">Quelltext bearbeiten</a>]</div> Ab Grad 8 treten bei vielen Newton-Cotes-Formeln negative Gewichte auf, was die Gefahr der <a href="/wiki/Ausl%C3%B6schung_(numerische_Mathematik)" title="Auslöschung (numerische Mathematik)">Auslöschung</a> mit sich bringt. Außerdem kann man im Allgemeinen keine Konvergenz erwarten, da die <a href="/wiki/Polynominterpolation" title="Polynominterpolation">Polynominterpolation</a> schlecht konditioniert ist. Bei größeren Integrationsbereichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}"> unterteilt man diese daher in einzelne Teilintervalle und wendet auf jedes einzelne Teilintervall eine Formel niedriger Ordnung an. <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=7" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Literatur">Quelltext bearbeiten</a>]</div> <ul><li>Hans R. Schwarz, Norbert Köckler: Numerische Mathematik. 6. Auflage. Teubner, Stuttgart 2006, <a href="/wiki/Spezial:ISBN-Suche/3519429608" class="internal mw-magiclink-isbn">ISBN 3-519-42960-8</a>, S. 311–316.</li> <li>Roland W. Freund, Ronald H. W. Hoppe: Stoer/Bulirsch: Numerische Mathematik 1. 10. Auflage. Springer, Berlin 2007, <a href="/wiki/Spezial:ISBN-Suche/9783540453895" class="internal mw-magiclink-isbn">ISBN 978-3-540-45389-5</a>, S. 164–169.</li> <li>Michael R. Schäferkotter, Prem K. Kythe: Handbook of Computational Methods for Integration. Chapman & Hall, Boca Raton 2005, <a href="/wiki/Spezial:ISBN-Suche/1584884282" class="internal mw-magiclink-isbn">ISBN 1-58488-428-2</a>, S. 54–62, 503–505.</li> <li>Günter Bärwolf: Numerik für Ingenieure, Physiker und Informatiker. <a href="/wiki/Spezial:ISBN-Suche/9783827416896" class="internal mw-magiclink-isbn">ISBN 978-3-8274-1689-6</a>, Spektrum, München 2007, S. 128.</li> <li><a href="/wiki/Gisela_Engeln-M%C3%BCllges" title="Gisela Engeln-Müllges">Gisela Engeln-Müllges</a>, Klaus Niederdrenk, Reinhard Wodicka: Numerik-Algorithmen : Verfahren, Beispiele, Anwendungen. <a href="/wiki/Spezial:ISBN-Suche/9783642134722" class="internal mw-magiclink-isbn">ISBN 978-3-642-13472-2</a>, Springer, Berlin und Heidelberg 2011.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2>[<a href="/w/index.php?title=Newton-Cotes-Formeln&veaction=edit&section=8" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor">Bearbeiten</a> | <a href="/w/index.php?title=Newton-Cotes-Formeln&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise">Quelltext bearbeiten</a>]</div> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> Thomas Weddle (<a href="/wiki/Newcastle-upon-Tyne" class="mw-redirect" title="Newcastle-upon-Tyne">Newcastle-upon-Tyne</a>): <cite style="font-style:italic">A new simple and general method of solving numerical equations of all orders</cite>. Hamilton, Adams & Co. and J. Philipson, London 1842 (<a rel="nofollow" class="external text" href="https://archive.org/details/newsimplegeneral00weddrich">Internet Archive</a> – 52 S.).<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:Newton-Cotes-Formeln&rft.au=Thomas+Weddle+%28Newcastle-upon-Tyne%29&rft.btitle=A+new+simple+and+general+method+of+solving+numerical+equations+of+all+orders&rft.date=1842&rft.genre=book&rft.place=London&rft.pub=Hamilton%2C+Adams+%26+Co.+and+J.+Philipson" style="display:none">  </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <a rel="nofollow" class="external text" href="https://www.wolframalpha.com/input/?i=Table%5BTable%5BIntegrate%5BProduct%5B(x-(j-1)/(m-1))/((i-j)/(m-1)),%7Bj,1,i-1%7D%5D*Product%5B(x-(j-1)/(m-1))/((i-j)/(m-1)),%7Bj,i%2B1,m%7D%5D,%7Bx,0,1%7D%5D+,%7Bi,1,m%7D%5D,%7Bm,2,9%7D%5D">WolframAlpha.</a> wolframalpha.com, abgerufen am 14. September 2019.<span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3ANewton-Cotes-Formeln&rft.title=WolframAlpha&rft.description=WolframAlpha&rft.identifier=&rft.publisher=wolframalpha.com&rft.date=">  </li> </ol></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Newton-Cotes-Formeln&oldid=247880104">https://de.wikipedia.org/w/index.php?title=Newton-Cotes-Formeln&oldid=247880104</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Numerische_Mathematik" title="Kategorie:Numerische Mathematik">Numerische Mathematik</a></li><li><a href="/wiki/Kategorie:Isaac_Newton_als_Namensgeber" title="Kategorie:Isaac Newton als Namensgeber">Isaac Newton als Namensgeber</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Versteckte Kategorie: <ul><li><a href="/wiki/Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Cite_web/tempor%C3%A4r" title="Kategorie:Wikipedia:Vorlagenfehler/Vorlage:Cite web/temporär">Wikipedia:Vorlagenfehler/Vorlage:Cite web/temporär</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > Meine Werkzeuge </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item">Nicht angemeldet</li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n">Diskussionsseite</a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y">Beiträge</a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Newton-Cotes-Formeln" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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Newton-Cotes-Formeln – Wikipedia