查看“︁指数函数积分表”︁的源代码

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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)

: <math>\int e^{cx}\;dx = \frac{1}{c} e^{cx}</math>

: <math>\int a^{cx}\;dx = \frac{1}{c \ln a} a^{cx} \qquad\qquad\mbox{(} a > 0,\mbox{ }a \ne 1\mbox{)}</math>

: <math>\int xe^{cx}\; dx = \frac{e^{cx}}{c^2}(cx-1)</math>

: <math>\int x^2 e^{cx}\;dx = e^{cx}\left(\frac{x^2}{c}-\frac{2x}{c^2}+\frac{2}{c^3}\right)</math>

: <math>\int x^n e^{cx}\; dx = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} dx</math>

: <math>\int\frac{e^{cx}\; dx}{x} = \ln|x| +\sum_{i=1}^\infty\frac{(cx)^i}{i\cdot i!}</math>

: <math>\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{)}</math>

: <math>\int e^{cx}\ln x\; dx = \frac{1}{c}e^{cx}\ln|x|-\operatorname{Ei}\,(cx)</math>

: <math>\int e^{cx}\sin bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx)</math>

: <math>\int e^{cx}\cos bx\; dx = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx)</math>

: <math>\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</math>

: <math>\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</math>

:<math>\int x e^{c x^2 }\; dx= \frac{1}{2c} \;  e^{c x^2}</math>

:<math>\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)</math>

:<math>\int e^{x^2}\,dx =\sum_{n=0}^{\infty} \frac{x^{2n+1}}{n!(2n+1)}</math>

:<math>\int_{-\infty}^{\infty} e^{-ax^2}\,dx=\sqrt{\pi \over a}</math> （[[高斯积分]]）

:<math>\int_{0}^{\infty} x^{2n} e^{-\frac{x^2}{a^2}}\,dx=\sqrt{\pi} {(2n)! \over {n!}} {\left (\frac{a}{2} \right)}^{2n + 1}</math>

{{Lists of integrals}}

[[Category:积分表]]
[[Category:指数]]