Benutzer:Patrick2000/FuT

Sonstige

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

Trigonometrische und Hyperbelfunktionen

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

Integrationsregeln

$\int (f+g)=\int f+\int g$

$\int uv'=uv-\int u'v$

$\int _{a}^{b}(f\circ g)g'=\int _{g(a)}^{g(b)}f$