## Encyclopedia > List of integrals of exponential functions

Article Content

# 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.

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

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

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

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

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

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

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

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

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

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

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

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Springs, New York ... Springs is located at 41°1'31" North, 72°9'31" West (41.025238, -72.158499)1. According to the United States Census Bureau, the town has a total area of 23.9 ...