List of integrals of exponential functions

The following is a list of integrals (antiderivative functions) of exponential functions. For a complete list of Integral functions, please see table of integrals and list of integrals.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\int e^{x^2}\,dx = e^{x^2}\left (\sum_{r=1}^n\frac{1}{2^n x^{2n-1}} \right)+ \frac {2n-1}{2^n}\int \frac{e^{x^2}\;dx}{x^{2n}}$

$\int_{-\infty}^{\infty} e^{-ax^2}\,dx=\sqrt{\pi \over a}$

See also: List of integrals of exponential functions, Antiderivative, Exponential function, Integral, List of integrals, Table of integrals