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# Rechenregeln zum Integral

Rechenregeln:

${\int }_{a\phantom{\rule{0.12em}{0ex}}}^{b\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}=-{\int }_{b\phantom{\rule{0.12em}{0ex}}}^{a\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

Damit folgt sofort: ${\int }_{a\phantom{\rule{0.12em}{0ex}}}^{a\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}=0$

Beispiel

denn:

${\int }_{0}^{1}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{0}^{1}=1-0=-\left(0-1\right)=-{\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{1}^{0}=-{\int }_{1}^{0}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

${\int }_{a\phantom{\rule{0.12em}{0ex}}}^{b\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}+{\int }_{b\phantom{\rule{0.12em}{0ex}}}^{c\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\int }_{a\phantom{\rule{0.12em}{0ex}}}^{c\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

Beispiel

denn:

${\int }_{0}^{1}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}+{\int }_{1}^{2}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{0}^{1}+{\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{1}^{2}=\left(1-0\right)+\left(4-1\right)=4-0={\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{0}^{2}={\int }_{0}^{2}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

Konstante

Beispiel

denn:

$3\cdot {\int }_{1}^{2}{x\phantom{\rule{0.12em}{0ex}}}^{2}=3\cdot {\left[\frac{1}{3}{x\phantom{\rule{0.12em}{0ex}}}^{3}\right]}_{1}^{2}=3\cdot \left(\frac{1}{3}\cdot 8-\frac{1}{3}\right)=8-1={\left[{x\phantom{\rule{0.12em}{0ex}}}^{3}\right]}_{1}^{2}={\int }_{1}^{2}3\cdot {x\phantom{\rule{0.12em}{0ex}}}^{2}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

${\int }_{a\phantom{\rule{0.12em}{0ex}}}^{b\phantom{\rule{0.12em}{0ex}}}\left(f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)±g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\int }_{a\phantom{\rule{0.12em}{0ex}}}^{b\phantom{\rule{0.12em}{0ex}}}f\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}±{\int }_{a\phantom{\rule{0.12em}{0ex}}}^{b\phantom{\rule{0.12em}{0ex}}}g\phantom{\rule{0.12em}{0ex}}\left(x\phantom{\rule{0.12em}{0ex}}\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

Beispiel

denn:

${\int }_{0}^{1}\left(2x\phantom{\rule{0.12em}{0ex}}+1\right)d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}+x\phantom{\rule{0.12em}{0ex}}\right]}_{0}^{1}=\left({1}^{2}+1\right)-\left({0}^{2}+0\right)=\left({1}^{2}-{0}^{2}\right)+\left(1-0\right)={\left[{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{0}^{1}+{\left[x\phantom{\rule{0.12em}{0ex}}\right]}_{0}^{1}={\int }_{0}^{1}2x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}+{\int }_{0}^{1}1d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}$

Beispiel

denn:

und

${\int }_{-1}^{1}|x\phantom{\rule{0.12em}{0ex}}|d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\int }_{-1}^{0}-x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}+{\int }_{0}^{1}x\phantom{\rule{0.12em}{0ex}}d\phantom{\rule{0.12em}{0ex}}x\phantom{\rule{0.12em}{0ex}}={\left[-\frac{1}{2}{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{-1}^{0}+{\left[\frac{1}{2}{x\phantom{\rule{0.12em}{0ex}}}^{2}\right]}_{0}^{1}=\left(0-\left(-\frac{1}{2}\right)\right)+\left(\frac{1}{2}+0\right)=\frac{1}{2}+\frac{1}{2}=1$

$⇒0<1$

## Verknüpfte Inhalte

Kategorie: Bestimmtes Integral

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