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.
- <math>\int e^{cx} dx = \frac{1}{c} e^{cx}</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} dx}{x^{n-1}}\right) \qquad\mbox{(for }n\neq 1\mbox{)}</math>
- <math>\int e^{cx}\ln x\; dx = \frac{1}{c}\left(e^{cx}\ln|x|-\int\frac{e^{cx} dx}{x}\right)</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>