Formelsammlung Mathematik: Unbestimmte Integrale exponentieller Funktionen

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Nachfolgende Liste enthält einige Integrale exponentieller Funktionen

$\int e^{cx}\;dx={\frac {1}{c}}e^{cx}$

$\int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(}}a>0,{\mbox{ }}a\neq 1{\mbox{)}}$

$\int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)$

$\int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)$

$\int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx=e^{cx}\left(\sum _{i=0}^{n}\;(-1)^{i}\;c^{-i-1}{\frac {n!}{(n-i)!}}\;x^{n-i}\right)$

$\int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}$

$\int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}$

$\int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)$

$\int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)$

$\int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)$

$\int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx$

$\int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx$

$\int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}$

$\int x^{n-1}e^{cx^{n}}\;dx={\frac {1}{nc}}\;e^{cx^{n}}\qquad {\mbox{(}}n>1,{\mbox{ }}n\in \mathbb {N} {\mbox{)}}$

$\int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})$

$\int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;dx\quad {\mbox{wenn }}n>0,$

wobei $c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {2j\,!}{j!\,2^{2j+1}}}\ .$

$\int {_{\underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} } \atop _{m}}dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(für }}x>0{\mbox{)}}$

wobei $a_{mn}={\begin{cases}1&{\text{wenn }}n=0,\\{\frac {1}{n!}}&{\text{wenn }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{sonst}}\end{cases}}$

$\int _{-\infty }^{\infty }e^{-ax^{2}}\,dx={\sqrt {\pi \over a}}$ (das Gauß'sche Fehlerintegral)

$\int _{-\infty }^{\infty }e^{-ax^{2}}e^{bx}\,dx={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{4a}}$

$\int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,dx=b{\sqrt {\pi \over a}}$

$\int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx=2\int _{0}^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a^{3}}}}$

$\int _{0}^{\infty }x^{2n}e^{-{x^{2}}/{a^{2}}}\,dx={\sqrt {\pi }}{(2n)! \over {n!}}{\left({\frac {a}{2}}\right)}^{2n+1}$

$\int _{0}^{2\pi }e^{x\cos \vartheta }d\vartheta =2\pi I_{0}(x)$ ($I_{0}$ ist die modifizierte Besselfunktion erster Ordnung)

$\int _{0}^{2\pi }e^{x\cos \vartheta +y\sin \vartheta }d\vartheta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)$

$\int _{0}^{\infty }x^{a}e^{-bx}dx={\frac {a!}{b^{a+1}}}$

Siehe auch

• Englische Wikipedia