List of integrals of exponential functions

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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[edit]

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.

\int e^{x}\;\mathrm{d}x = e^{x}
\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} for a > 0,\ a \ne 1
\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}
\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{)}
\int e^{cx}\ln x\; \mathrm{d}x = \frac{1}{c}\left(e^{cx}\ln|x|-\operatorname{Ei}\,(cx)\right)
\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
\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)
\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 } n > 0,
where  c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{(2j)\,!}{j!\, 2^{2j+1}} \ .
 {\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[edit]


\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[edit]