指数函数积分表

以下是部分指數函數的積分表(书写时省略了不定积分结果中都含有的任意常数Cn)

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

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

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

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

${\displaystyle \int x^{n}e^{cx}\;dx={\frac {1}{c}x^{n}e^{cx}-{\frac {n}{c}\int x^{n-1}e^{cx}dx}$

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

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

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

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

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

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

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

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

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

${\displaystyle \int e^{x^{2}\,dx=\sum _{n=0}^{\infty }{\frac {x^{2n+1}{n!(2n+1)}$

${\displaystyle \int _{-\infty }^{\infty }e^{-ax^{2}\,dx={\sqrt {\pi \over a}$ （高斯积分）

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