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Integrale exponentieller Funktionen
∫
e
c
x
d
x
=
1
c
e
c
x
{\displaystyle \int e^{cx}\;dx={\frac {1}{c}}e^{cx}}
∫
a
c
x
d
x
=
1
c
ln
a
a
c
x
(
a
>
0
,
a
≠
1
)
{\displaystyle \int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(}}a>0,{\mbox{ }}a\neq 1{\mbox{)}}}
∫
x
e
c
x
d
x
=
e
c
x
c
2
(
c
x
−
1
)
{\displaystyle \int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)}
∫
x
2
e
c
x
d
x
=
e
c
x
(
x
2
c
−
2
x
c
2
+
2
c
3
)
{\displaystyle \int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)}
∫
x
n
e
c
x
d
x
=
1
c
x
n
e
c
x
−
n
c
∫
x
n
−
1
e
c
x
d
x
{\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx}
∫
e
c
x
d
x
x
=
ln
|
x
|
+
∑
i
=
1
∞
(
c
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}}
∫
e
c
x
d
x
x
n
=
1
n
−
1
(
−
e
c
x
x
n
−
1
+
c
∫
e
c
x
x
n
−
1
d
x
)
(
n
≠
1
)
{\displaystyle \int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}}
∫
e
c
x
ln
x
d
x
=
1
c
e
c
x
ln
|
x
|
−
Ei
(
c
x
)
{\displaystyle \int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)}
∫
e
c
x
sin
b
x
d
x
=
e
c
x
c
2
+
b
2
(
c
sin
b
x
−
b
cos
b
x
)
{\displaystyle \int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)}
∫
e
c
x
cos
b
x
d
x
=
e
c
x
c
2
+
b
2
(
c
cos
b
x
+
b
sin
b
x
)
{\displaystyle \int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)}
∫
e
c
x
sin
n
x
d
x
=
e
c
x
sin
n
−
1
x
c
2
+
n
2
(
c
sin
x
−
n
cos
x
)
+
n
(
n
−
1
)
c
2
+
n
2
∫
e
c
x
sin
n
−
2
x
d
x
{\displaystyle \int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx}
∫
e
c
x
cos
n
x
d
x
=
e
c
x
cos
n
−
1
x
c
2
+
n
2
(
c
cos
x
+
n
sin
x
)
+
n
(
n
−
1
)
c
2
+
n
2
∫
e
c
x
cos
n
−
2
x
d
x
{\displaystyle \int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx}
∫
x
e
c
x
2
d
x
=
1
2
c
e
c
x
2
{\displaystyle \int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}}
∫
x
n
−
1
e
c
x
n
d
x
=
1
n
c
e
c
x
n
(
n
>
1
,
n
∈
N
)
{\displaystyle \int x^{n-1}e^{cx^{n}}\;dx={\frac {1}{nc}}\;e^{cx^{n}}\qquad {\mbox{(}}n>1,{\mbox{ }}n\in \mathbb {N} {\mbox{)}}}
∫
1
σ
2
π
e
−
(
x
−
μ
)
2
/
2
σ
2
d
x
=
1
2
σ
(
1
+
erf
x
−
μ
σ
2
)
{\displaystyle \int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})}
∫
e
x
2
d
x
=
e
x
2
(
∑
j
=
0
n
−
1
c
2
j
1
x
2
j
+
1
)
+
(
2
n
−
1
)
c
2
n
−
2
∫
e
x
2
x
2
n
d
x
wenn
n
>
0
,
{\displaystyle \int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;dx\quad {\mbox{wenn }}n>0,}
wobei
c
2
j
=
1
⋅
3
⋅
5
⋯
(
2
j
−
1
)
2
j
+
1
=
2
j
!
j
!
2
2
j
+
1
.
{\displaystyle c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {2j\,!}{j!\,2^{2j+1}}}\ .}
∫
x
x
⋅
⋅
x
⏟
m
d
x
=
∑
n
=
0
m
(
−
1
)
n
(
n
+
1
)
n
−
1
n
!
Γ
(
n
+
1
,
−
ln
x
)
+
∑
n
=
m
+
1
∞
(
−
1
)
n
a
m
n
Γ
(
n
+
1
,
−
ln
x
)
(für
x
>
0
)
{\displaystyle \int {_{\underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} } \atop _{m}}dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(für }}x>0{\mbox{)}}}
wobei
a
m
n
=
{
1
wenn
n
=
0
,
1
n
!
wenn
m
=
1
,
1
n
∑
j
=
1
n
j
a
m
,
n
−
j
a
m
−
1
,
j
−
1
sonst
{\displaystyle a_{mn}={\begin{cases}1&{\text{wenn }}n=0,\\{\frac {1}{n!}}&{\text{wenn }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{sonst}}\end{cases}}}
∫
−
∞
∞
e
−
a
x
2
d
x
=
π
a
{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}\,dx={\sqrt {\pi \over a}}}
(das
Gauß'sche Fehlerintegral
)
∫
−
∞
∞
e
−
a
x
2
e
b
x
d
x
=
π
a
e
b
2
4
a
{\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}}e^{bx}\,dx={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{4a}}}
∫
−
∞
∞
x
e
−
a
(
x
−
b
)
2
d
x
=
b
π
a
{\displaystyle \int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,dx=b{\sqrt {\pi \over a}}}
∫
−
∞
∞
x
2
e
−
a
x
2
d
x
=
2
∫
0
∞
x
2
e
−
a
x
2
d
x
=
1
2
π
a
3
{\displaystyle \int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx=2\int _{0}^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a^{3}}}}}
∫
0
∞
x
2
n
e
−
x
2
/
a
2
d
x
=
π
(
2
n
)
!
n
!
(
a
2
)
2
n
+
1
{\displaystyle \int _{0}^{\infty }x^{2n}e^{-{x^{2}}/{a^{2}}}\,dx={\sqrt {\pi }}{(2n)! \over {n!}}{\left({\frac {a}{2}}\right)}^{2n+1}}
∫
0
2
π
e
x
cos
ϑ
d
ϑ
=
2
π
I
0
(
x
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \vartheta }d\vartheta =2\pi I_{0}(x)}
(
I
0
{\displaystyle I_{0}}
ist die modifizierte
Besselfunktion
erster Ordnung)
∫
0
2
π
e
x
cos
ϑ
+
y
sin
ϑ
d
ϑ
=
2
π
I
0
(
x
2
+
y
2
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \vartheta +y\sin \vartheta }d\vartheta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}
∫
0
∞
x
a
e
−
b
x
d
x
=
a
!
b
a
+
1
{\displaystyle \int _{0}^{\infty }x^{a}e^{-bx}dx={\frac {a!}{b^{a+1}}}}
Integrale logarithmischer Funktionen
Bearbeiten
Hinweis
Es wird angenommen, dass
x
> 0 ist.
∫
ln
c
x
d
x
=
x
ln
c
x
−
x
{\displaystyle \int \ln cx\;dx=x\ln cx-x}
∫
ln
(
a
x
+
b
)
d
x
=
x
ln
(
a
x
+
b
)
−
x
+
b
a
ln
(
a
x
+
b
)
{\displaystyle \int \ln(ax+b)\;dx=x\ln(ax+b)-x+{\frac {b}{a}}\ln(ax+b)}
∫
(
ln
x
)
2
d
x
=
x
(
ln
x
)
2
−
2
x
ln
x
+
2
x
{\displaystyle \int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x}
∫
(
ln
c
x
)
n
d
x
=
x
(
ln
c
x
)
n
−
n
∫
(
ln
c
x
)
n
−
1
d
x
{\displaystyle \int (\ln cx)^{n}\;dx=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}dx}
∫
d
x
ln
x
=
ln
|
ln
x
|
+
ln
x
+
∑
i
=
2
∞
(
ln
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}}
∫
d
x
(
ln
x
)
n
=
−
x
(
n
−
1
)
(
ln
x
)
n
−
1
+
1
n
−
1
∫
d
x
(
ln
x
)
n
−
1
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}
∫
x
m
ln
x
d
x
=
x
m
+
1
(
ln
x
m
+
1
−
1
(
m
+
1
)
2
)
(
m
≠
−
1
)
{\displaystyle \int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(}}m\neq -1{\mbox{)}}}
∫
x
m
(
ln
x
)
n
d
x
=
x
m
+
1
(
ln
x
)
n
m
+
1
−
n
m
+
1
∫
x
m
(
ln
x
)
n
−
1
d
x
(
m
≠
−
1
)
{\displaystyle \int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(}}m\neq -1{\mbox{)}}}
∫
(
ln
x
)
n
d
x
x
=
(
ln
x
)
n
+
1
n
+
1
(
n
≠
−
1
)
{\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(}}n\neq -1{\mbox{)}}}
∫
ln
x
d
x
x
m
=
−
ln
x
(
m
−
1
)
x
m
−
1
−
1
(
m
−
1
)
2
x
m
−
1
(
m
≠
1
)
{\displaystyle \int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}}
∫
(
ln
x
)
n
d
x
x
m
=
−
(
ln
x
)
n
(
m
−
1
)
x
m
−
1
+
n
m
−
1
∫
(
ln
x
)
n
−
1
d
x
x
m
(
m
≠
1
)
{\displaystyle \int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}}
∫
x
m
d
x
(
ln
x
)
n
=
−
x
m
+
1
(
n
−
1
)
(
ln
x
)
n
−
1
+
m
+
1
n
−
1
∫
x
m
d
x
(
ln
x
)
n
−
1
(
n
≠
1
)
{\displaystyle \int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}
∫
d
x
x
ln
x
=
ln
|
ln
x
|
{\displaystyle \int {\frac {dx}{x\ln x}}=\ln |\ln x|}
∫
d
x
x
n
ln
x
=
ln
|
ln
x
|
+
∑
i
=
1
∞
(
−
1
)
i
(
n
−
1
)
i
(
ln
x
)
i
i
⋅
i
!
{\displaystyle \int {\frac {dx}{x^{n}\ln x}}=\ln |\ln x|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i!}}}
∫
d
x
x
(
ln
x
)
n
=
−
1
(
n
−
1
)
(
ln
x
)
n
−
1
(
n
≠
1
)
{\displaystyle \int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}}
∫
sin
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
−
cos
(
ln
x
)
)
{\displaystyle \int \sin(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))}
∫
cos
(
ln
x
)
d
x
=
x
2
(
sin
(
ln
x
)
+
cos
(
ln
x
)
)
{\displaystyle \int \cos(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))}
∫
e
x
(
x
ln
x
−
x
−
1
x
)
d
x
=
e
x
(
x
ln
x
−
x
−
ln
x
)
{\displaystyle \int e^{x}(x\ln x-x-{\frac {1}{x}})\;dx=e^{x}(x\ln x-x-\ln x)}
(das Gauß'sche Fehlerintegral)
(
ist die modifizierte Besselfunktion erster Ordnung)
