# Benutzer:Patrick2000/FuT

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### Sonstige

Funktion ${\displaystyle f(x)}$ Stammfunktion ${\displaystyle F(x)}$
${\displaystyle e^{-x^{2}}}$ ${\displaystyle {\frac {\sqrt {\pi }}{2}}\;\operatorname {Erf} \;x}$
${\displaystyle e^{-ax^{2}+bx+c}}$ ${\displaystyle {\frac {\sqrt {\pi }}{2{\sqrt {a}}}}\;e^{{\frac {b^{2}}{4a}}+c}\;\operatorname {Erf} \;\left({\sqrt {a}}\;x-{\frac {b}{2{\sqrt {a}}}}\right)}$
${\displaystyle {\frac {u'(x)}{u(x)}}}$ ${\displaystyle \ln \left|u(x)\right|\,}$
${\displaystyle u'(x)\cdot u(x)}$ ${\displaystyle {\tfrac {1}{2}}(u(x))^{2}}$

### Trigonometrische und Hyperbelfunktionen

Funktion ${\displaystyle f(x)}$ Stammfunktion ${\displaystyle F(x)}$
${\displaystyle \sin x\;}$ ${\displaystyle -\cos x\;}$
${\displaystyle \cos x\;}$ ${\displaystyle \sin x\;}$
${\displaystyle \sin ^{2}x\;}$ ${\displaystyle {\tfrac {1}{2}}(x-\sin x\cdot \cos x)\;}$
${\displaystyle \cos ^{2}x\;}$ ${\displaystyle {\tfrac {1}{2}}(x+\sin x\cdot \cos x)\;}$
${\displaystyle \sin ax\cos ax\;}$ ${\displaystyle -{\frac {1}{2a}}\cos ^{2}ax\,\!}$
${\displaystyle \tan x\;}$ ${\displaystyle -\ln |\cos x|\;}$
${\displaystyle \cot x\;}$ ${\displaystyle \ln |\sin x|\;}$
${\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x\;}$ ${\displaystyle \tan x\;}$
${\displaystyle {\frac {-1}{\sin ^{2}x}}=-(1+\cot ^{2}x)\;}$ ${\displaystyle \cot x\;}$
${\displaystyle \arcsin x\;}$ ${\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}\;}$
${\displaystyle \arccos x\;}$ ${\displaystyle x\arccos x-{\sqrt {1-x^{2}}}\;}$
${\displaystyle \arctan x\;}$ ${\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)\;}$
${\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}\;}$ ${\displaystyle \arcsin x\;}$
${\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}\;}$ ${\displaystyle \arccos x\;}$
${\displaystyle {\frac {1}{x^{2}+1}}\;}$ ${\displaystyle \arctan x\;}$
${\displaystyle {\frac {x^{2}}{x^{2}+1}}\;}$ ${\displaystyle x-\arctan x\;}$
${\displaystyle {\frac {1}{(x^{2}+1)^{2}}}\;}$ ${\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)\;}$
${\displaystyle \sinh x\;}$ ${\displaystyle \cosh x\;}$
${\displaystyle \cosh x\;}$ ${\displaystyle \sinh x\;}$
${\displaystyle \tanh x\;}$ ${\displaystyle \ln \cosh x\;}$
${\displaystyle \coth x\;}$ ${\displaystyle \ln |\sinh x|\;}$
${\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x\;}$ ${\displaystyle \tanh x\;}$
${\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x\;}$ ${\displaystyle \coth x\;}$
${\displaystyle \operatorname {arsinh} \;x\;}$ ${\displaystyle x\;\operatorname {arsinh} \;x-{\sqrt {x^{2}+1}}\;}$
${\displaystyle \operatorname {arcosh} \;x\;}$ ${\displaystyle x\;\operatorname {arcosh} \;x-{\sqrt {x^{2}-1}}\;}$
${\displaystyle \operatorname {artanh} \;x\;}$ ${\displaystyle x\;\operatorname {artanh} \;x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}\;}$
${\displaystyle \operatorname {arcoth} \;x\;}$ ${\displaystyle x\;\operatorname {arcoth} \;x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}\;}$
${\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}\;}$ ${\displaystyle \operatorname {arsinh} \;x\;}$
${\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\;,\;x>1}$ ${\displaystyle \operatorname {arcosh} \;x\;}$
${\displaystyle {\frac {1}{1-x^{2}}}\;,\;\left|x\right|<1}$ ${\displaystyle \operatorname {artanh} \;x\;}$
${\displaystyle {\frac {1}{1-x^{2}}}\;,\;\left|x\right|>1}$ ${\displaystyle \operatorname {arcoth} \;x\;}$
${\displaystyle \sin ^{2}kx\;}$ ${\displaystyle {\frac {x}{2}}-{\frac {\sin(2kx)}{4k}}}$
${\displaystyle \cos ^{2}kx\;}$ ${\displaystyle {\frac {x}{2}}+{\frac {\sin(2kx)}{4k}}}$

### Integrationsregeln

${\displaystyle \int (f+g)=\int f+\int g}$
${\displaystyle \int uv'=uv-\int u'v}$
${\displaystyle \int _{a}^{b}(f\circ g)g'=\int _{g(a)}^{g(b)}f}$