List of integrals of exponential functions

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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} }{x^{n-1}}\,dx\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_{j=0}^{n-1}c_{2j}\,\frac{1}{x^{2j+1}} \right )+(2n-1)c_{2n-2} \int \frac{e^{x^2}}{x^{2n}}\;dx \quad \mbox{valid for } n > 0,
where c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{2j\,!}{j!\, 2^{2j+1}} \ .
\int_{-\infty}^{\infty} e^{-ax^2}\,dx=\sqrt{\pi \over a} (the Gaussian integral)
\int_{-\infty}^{\infty} e^{-ax^2} e^{bx}\,dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}}
\int_{-\infty}^{\infty} x e^{-a(x-b)^2}\,dx=b \sqrt{\pi \over a}
\int_{-\infty}^{\infty} x^2 e^{-ax^2}\,dx=\frac{1}{2} \sqrt{\pi \over a^3}
\int_{0}^{\infty} x^{2n} e^{-{x^2}/{a^2}}\,dx=\sqrt{\pi} {(2n)! \over {n!}} {\left (\frac{a}{2} \right)}^{2n + 1}
\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x) (I0 is the modified Bessel function of the first kind)
\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left( \sqrt{x^2 + y^2} \right)
\int_{0}^{\infty} x^a e^{-bx} dx = \frac{a!}{b^{a+1}}
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