List of integrals of exponential functions

The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals.

Indefinite integral

Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity.

Integrals involving only exponential functions

$\int f'(x)e^{f(x)}\;\mathrm{d}x = e^{f(x)}$

$\int e^{cx}\;\mathrm{d}x = \frac{1}{c} e^{cx}$

$\int a^{cx}\;\mathrm{d}x = \frac{1}{c\cdot \ln a} a^{cx}$f for $a > 0,\ a \ne 1$

Integrals involving exponential and power functions

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

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

$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n (-1)^i\,\frac{n!}{(n-i)!\,c^{i+1}}\,x^{n-i} = e^{cx}\sum_{i=0}^n (-1)^{n-i}\,\frac{n!}{i!\,c^{n-i+1}}\,x^i$

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

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

Integrals involving exponential and trigonometric functions

$\int e^{cx}\sin bx\; \mathrm{d}x = \frac{e^{cx}}{c^2+b^2}(c\sin bx - b\cos bx) = \frac{e^{cx}}{\sqrt{c^2+b^2}}\sin(bx-\phi)\qquad \cos(\phi) = \frac{c}{\sqrt{c^2+b^2}}$

$\int e^{cx}\cos bx\; \mathrm{d}x = \frac{e^{cx}}{c^2+b^2}(c\cos bx + b\sin bx) = \frac{e^{cx}}{\sqrt{c^2+b^2}}\cos(bx-\phi)\qquad \cos(\phi) = \frac{c}{\sqrt{c^2+b^2}}$

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

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

Integrals involving the error function

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

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

$\int e^{-c x^2 }\; \mathrm{d}x= \sqrt{\frac{\pi}{4c}} \operatorname{erf}(\sqrt{c} x)$ ($\operatorname{erf}$ is the error function)

$\int xe^{-c x^2 }\; \mathrm{d}x=-\frac{1}{2c}e^{-cx^2}$

$\int\frac{e^{-x^2}}{x^2}\; \mathrm{d}x = -\frac{e^{-x^2}}{x} - \sqrt{\pi} \mathrm{erf} (x)$

$\int {\frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }}\; \mathrm{d}x= \frac{1}{2} \left(\operatorname{erf}\,\frac{x-\mu}{\sigma \sqrt{2}}\right)$

Other integrals

$\int e^{x^2}\,\mathrm{d}x = e^{x^2}\left( \sum_{j=0}^{n-1}c_{2j}\,\frac{1}{x^{2j+1}} \right )+(2n-1)c_{2n-2} \int \frac{e^{x^2}}{x^{2n}}\;\mathrm{d}x \quad \mbox{valid for any } n > 0,$

where $c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{(2j)\,!}{j!\, 2^{2j+1}} \ .$

(Note that the value of the expression is independent of the value of $n$, which is why it does not appear in the integral.)

${\int \underbrace{x^{x^{\cdot^{\cdot^{x}}}}}_m \,dx= \sum_{n=0}^m\frac{(-1)^n(n+1)^{n-1}}{n!}\Gamma(n+1,- \ln x) + \sum_{n=m+1}^\infty(-1)^na_{mn}\Gamma(n+1,-\ln x) \qquad\mbox{(for }x> 0\mbox{)}}$

where $a_{mn}=\begin{cases}1 &\text{if } n = 0, \\ \frac{1}{n!} &\text{if } m=1, \\ \frac{1}{n}\sum_{j=1}^{n}ja_{m,n-j}a_{m-1,j-1} &\text{otherwise} \end{cases}$

and $\Gamma(x,y)$ is the gamma function

$\int \frac{1}{ae^{\lambda x} + b} \; \mathrm{d}x = \frac{x}{b} - \frac{1}{b \lambda} \ln\left(a e^{\lambda x} + b \right) \,$ when $b \neq 0$, $\lambda \neq 0$, and $ae^{\lambda x} + b > 0 \,.$

$\int \frac{e^{2\lambda x}}{ae^{\lambda x} + b} \; \mathrm{d}x = \frac{1}{a^2 \lambda} \left[a e^{\lambda x} + b - b \ln\left(a e^{\lambda x} + b \right) \right] \,$ when $a \neq 0$, $\lambda \neq 0$, and $ae^{\lambda x} + b > 0 \,.$

Definite integrals

$\int_0^1 e^{x\cdot \ln a + (1-x)\cdot \ln b}\;\mathrm{d}x = \int_0^1 \left(\frac{a}{b}\right)^{x}\cdot b\;\mathrm{d}x = \int_0^1 a^{x}\cdot b^{1-x}\;\mathrm{d}x = \frac{a-b}{\ln a - \ln b}$ for $a > 0,\ b > 0,\ a \ne b$, which is the logarithmic mean

$\int_{0}^{\infty} e^{-ax}\,\mathrm{d}x=\frac{1}{a} \quad (\operatorname{Re}(a)>0)$

$\int_{0}^{\infty} e^{-ax^2}\,\mathrm{d}x=\frac{1}{2} \sqrt{\pi \over a} \quad (a>0)$ (the Gaussian integral)

$\int_{-\infty}^{\infty} e^{-ax^2}\,\mathrm{d}x=\sqrt{\pi \over a} \quad (a>0)$

$\int_{-\infty}^{\infty} e^{-ax^2} e^{-2bx}\,\mathrm{d}x=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}} \quad (a>0)$ (see Integral of a Gaussian function)

$\int_{-\infty}^{\infty} x e^{-a(x-b)^2}\,\mathrm{d}x= b \sqrt{\frac{\pi}{a}} \quad (\operatorname{Re}(a)>0)$

$\int_{-\infty}^{\infty} x e^{-ax^2+bx}\,\mathrm{d}x= \frac{ \sqrt{\pi} b }{2a^{3/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)$

$\int_{-\infty}^{\infty} x^2 e^{-ax^2}\,\mathrm{d}x=\frac{1}{2} \sqrt{\pi \over a^3} \quad (a>0)$

$\int_{-\infty}^{\infty} x^2 e^{-ax^2-bx}\,\mathrm{d}x=\frac{\sqrt{\pi}(2a+b^2)}{4a^{5/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)$

$\int_{-\infty}^{\infty} x^3 e^{-ax^2+bx}\,\mathrm{d}x=\frac{\sqrt{\pi}(6a+b^2)b}{8a^{7/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)$

$\int_{0}^{\infty} x^{n} e^{-ax^2}\,\mathrm{d}x = \begin{cases} \frac{1}{2}\Gamma \left(\frac{n+1}{2}\right)/a^{\frac{n+1}{2}} & (n>-1,a>0) \\ \frac{(2k-1)!!}{2^{k+1}a^k}\sqrt{\frac{\pi}{a}} & (n=2k, k \;\text{integer}, a>0) \\ \frac{k!}{2a^{k+1}} & (n=2k+1,k \;\text{integer}, a>0) \end{cases}$ (!! is the double factorial)

$\int_{0}^{\infty} x^n e^{-ax}\,\mathrm{d}x = \begin{cases} \frac{\Gamma(n+1)}{a^{n+1}} & (n>-1,a>0) \\ \frac{n!}{a^{n+1}} & (n=0,1,2,\ldots,a>0) \\ \end{cases}$

$\int_{0}^{1} x^n e^{-ax}\,\mathrm{d}x = \frac{n!}{a^{n+1}}\left[ 1-e^{-a}\sum_{i=0}^{n} \frac{a^i}{i!} \right]$

$\int_0^\infty e^{-ax^b} dx = \frac{1}{b}\ a^{-\frac{1}{b}} \, \Gamma\left(\frac{1}{b}\right)$

$\int_0^\infty x^n e^{-ax^b} dx = \frac{1}{b}\ a^{-\frac{n+1}{b}} \, \Gamma\left(\frac{n+1}{b}\right)$

$\int_{0}^{\infty} e^{-ax}\sin bx \, \mathrm{d}x = \frac{b}{a^2+b^2} \quad (a>0)$

$\int_{0}^{\infty} e^{-ax}\cos bx \, \mathrm{d}x = \frac{a}{a^2+b^2} \quad (a>0)$

$\int_{0}^{\infty} xe^{-ax}\sin bx \, \mathrm{d}x = \frac{2ab}{(a^2+b^2)^2} \quad (a>0)$

$\int_{0}^{\infty} xe^{-ax}\cos bx \, \mathrm{d}x = \frac{a^2-b^2}{(a^2+b^2)^2} \quad (a>0)$

$\int_{0}^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_{0}(x)$ ($I_{0}$ is the modified Bessel function of the first kind)

$\int_{0}^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_{0} \left( \sqrt{x^2 + y^2} \right)$

References

• Wolfram Mathematica Online Integrator
• V. H. Moll, The Integrals in Gradshteyn and Ryzhik