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Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
e
−
x
2
{\displaystyle e^{-x^{2}}}
π
2
Erf
x
{\displaystyle {\frac {\sqrt {\pi }}{2}}\;\operatorname {Erf} \;x}
e
−
a
x
2
+
b
x
+
c
{\displaystyle e^{-ax^{2}+bx+c}}
π
2
a
e
b
2
4
a
+
c
Erf
(
a
x
−
b
2
a
)
{\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)}
u
′
(
x
)
u
(
x
)
{\displaystyle {\frac {u'(x)}{u(x)}}}
ln
|
u
(
x
)
|
{\displaystyle \ln \left|u(x)\right|\,}
u
′
(
x
)
⋅
u
(
x
)
{\displaystyle u'(x)\cdot u(x)}
1
2
(
u
(
x
)
)
2
{\displaystyle {\tfrac {1}{2}}(u(x))^{2}}
Trigonometrische und Hyperbelfunktionen
Bearbeiten
Funktion
f
(
x
)
{\displaystyle f(x)}
Stammfunktion
F
(
x
)
{\displaystyle F(x)}
sin
x
{\displaystyle \sin x\;}
−
cos
x
{\displaystyle -\cos x\;}
cos
x
{\displaystyle \cos x\;}
sin
x
{\displaystyle \sin x\;}
sin
2
x
{\displaystyle \sin ^{2}x\;}
1
2
(
x
−
sin
x
⋅
cos
x
)
{\displaystyle {\tfrac {1}{2}}(x-\sin x\cdot \cos x)\;}
cos
2
x
{\displaystyle \cos ^{2}x\;}
1
2
(
x
+
sin
x
⋅
cos
x
)
{\displaystyle {\tfrac {1}{2}}(x+\sin x\cdot \cos x)\;}
sin
a
x
cos
a
x
{\displaystyle \sin ax\cos ax\;}
−
1
2
a
cos
2
a
x
{\displaystyle -{\frac {1}{2a}}\cos ^{2}ax\,\!}
tan
x
{\displaystyle \tan x\;}
−
ln
|
cos
x
|
{\displaystyle -\ln |\cos x|\;}
cot
x
{\displaystyle \cot x\;}
ln
|
sin
x
|
{\displaystyle \ln |\sin x|\;}
1
cos
2
x
=
1
+
tan
2
x
{\displaystyle {\frac {1}{\cos ^{2}x}}=1+\tan ^{2}x\;}
tan
x
{\displaystyle \tan x\;}
−
1
sin
2
x
=
−
(
1
+
cot
2
x
)
{\displaystyle {\frac {-1}{\sin ^{2}x}}=-(1+\cot ^{2}x)\;}
cot
x
{\displaystyle \cot x\;}
arcsin
x
{\displaystyle \arcsin x\;}
x
arcsin
x
+
1
−
x
2
{\displaystyle x\arcsin x+{\sqrt {1-x^{2}}}\;}
arccos
x
{\displaystyle \arccos x\;}
x
arccos
x
−
1
−
x
2
{\displaystyle x\arccos x-{\sqrt {1-x^{2}}}\;}
arctan
x
{\displaystyle \arctan x\;}
x
arctan
x
−
1
2
ln
(
1
+
x
2
)
{\displaystyle x\arctan x-{\tfrac {1}{2}}\ln \left(1+x^{2}\right)\;}
1
1
−
x
2
{\displaystyle {\frac {1}{\sqrt {1-x^{2}}}}\;}
arcsin
x
{\displaystyle \arcsin x\;}
−
1
1
−
x
2
{\displaystyle {\frac {-1}{\sqrt {1-x^{2}}}}\;}
arccos
x
{\displaystyle \arccos x\;}
1
x
2
+
1
{\displaystyle {\frac {1}{x^{2}+1}}\;}
arctan
x
{\displaystyle \arctan x\;}
x
2
x
2
+
1
{\displaystyle {\frac {x^{2}}{x^{2}+1}}\;}
x
−
arctan
x
{\displaystyle x-\arctan x\;}
1
(
x
2
+
1
)
2
{\displaystyle {\frac {1}{(x^{2}+1)^{2}}}\;}
1
2
(
x
x
2
+
1
+
arctan
x
)
{\displaystyle {\frac {1}{2}}\left({\frac {x}{x^{2}+1}}+\arctan x\right)\;}
sinh
x
{\displaystyle \sinh x\;}
cosh
x
{\displaystyle \cosh x\;}
cosh
x
{\displaystyle \cosh x\;}
sinh
x
{\displaystyle \sinh x\;}
tanh
x
{\displaystyle \tanh x\;}
ln
cosh
x
{\displaystyle \ln \cosh x\;}
coth
x
{\displaystyle \coth x\;}
ln
|
sinh
x
|
{\displaystyle \ln |\sinh x|\;}
1
cosh
2
x
=
1
−
tanh
2
x
{\displaystyle {\frac {1}{\cosh ^{2}x}}=1-\tanh ^{2}x\;}
tanh
x
{\displaystyle \tanh x\;}
−
1
sinh
2
x
=
1
−
coth
2
x
{\displaystyle {\frac {-1}{\sinh ^{2}x}}=1-\coth ^{2}x\;}
coth
x
{\displaystyle \coth x\;}
arsinh
x
{\displaystyle \operatorname {arsinh} \;x\;}
x
arsinh
x
−
x
2
+
1
{\displaystyle x\;\operatorname {arsinh} \;x-{\sqrt {x^{2}+1}}\;}
arcosh
x
{\displaystyle \operatorname {arcosh} \;x\;}
x
arcosh
x
−
x
2
−
1
{\displaystyle x\;\operatorname {arcosh} \;x-{\sqrt {x^{2}-1}}\;}
artanh
x
{\displaystyle \operatorname {artanh} \;x\;}
x
artanh
x
+
1
2
ln
(
1
−
x
2
)
{\displaystyle x\;\operatorname {artanh} \;x+{\frac {1}{2}}\ln {\left(1-x^{2}\right)}\;}
arcoth
x
{\displaystyle \operatorname {arcoth} \;x\;}
x
arcoth
x
+
1
2
ln
(
x
2
−
1
)
{\displaystyle x\;\operatorname {arcoth} \;x+{\frac {1}{2}}\ln {\left(x^{2}-1\right)}\;}
1
x
2
+
1
{\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}\;}
arsinh
x
{\displaystyle \operatorname {arsinh} \;x\;}
1
x
2
−
1
,
x
>
1
{\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\;,\;x>1}
arcosh
x
{\displaystyle \operatorname {arcosh} \;x\;}
1
1
−
x
2
,
|
x
|
<
1
{\displaystyle {\frac {1}{1-x^{2}}}\;,\;\left|x\right|<1}
artanh
x
{\displaystyle \operatorname {artanh} \;x\;}
1
1
−
x
2
,
|
x
|
>
1
{\displaystyle {\frac {1}{1-x^{2}}}\;,\;\left|x\right|>1}
arcoth
x
{\displaystyle \operatorname {arcoth} \;x\;}
sin
2
k
x
{\displaystyle \sin ^{2}kx\;}
x
2
−
sin
(
2
k
x
)
4
k
{\displaystyle {\frac {x}{2}}-{\frac {\sin(2kx)}{4k}}}
cos
2
k
x
{\displaystyle \cos ^{2}kx\;}
x
2
+
sin
(
2
k
x
)
4
k
{\displaystyle {\frac {x}{2}}+{\frac {\sin(2kx)}{4k}}}
Integrationsregeln
Bearbeiten
∫
(
f
+
g
)
=
∫
f
+
∫
g
{\displaystyle \int (f+g)=\int f+\int g}
∫
u
v
′
=
u
v
−
∫
u
′
v
{\displaystyle \int uv'=uv-\int u'v}
∫
a
b
(
f
∘
g
)
g
′
=
∫
g
(
a
)
g
(
b
)
f
{\displaystyle \int _{a}^{b}(f\circ g)g'=\int _{g(a)}^{g(b)}f}
