List of integrals of exponential functions

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.

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\int xe^{cx}\,dx = e^{cx}\left(\frac{cx-1}{c^{2}}\right) \qquad \text{ for } c \neq 0;

• $\backslash int\; x^2\; e^\{cx\}\backslash ,dx\; =\; e^\{cx\}\backslash left(\backslash frac\{x^2\}\{c\}-\backslash frac\{2x\}\{c^2\}+\backslash frac\{2\}\{c^3\}\backslash right)$
• $\backslash begin\{align\}$

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

&= \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

\end{align}

• $\backslash int\backslash frac\{e^\{cx\}\}\{x\}\backslash ,dx\; =\; \backslash ln|x|\; +\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{(cx)^n\}\{n\backslash cdot\; n!\}$
• $\backslash int\backslash frac\{e^\{cx\}\}\{x^n\}\backslash ,dx\; =\; \backslash frac\{1\}\{n-1\}\backslash left(-\backslash frac\{e^\{cx\}\}\{x^\{n-1\}\}+c\backslash int\backslash frac\{e^\{cx\}\; \}\{x^\{n-1\}\}\backslash ,dx\backslash right)\; \backslash qquad\backslash text\{(for\; \}n\backslash neq\; 1\backslash text\{)\}$

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• $\backslash int\; f\text{'}(x)e^\{f(x)\}\backslash ,dx\; =\; e^\{f(x)\}$
• $\backslash int\; e^\{cx\}\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\; e^\{cx\}$
• $\backslash int\; a^\{x\}\backslash ,dx\; =\; \backslash frac\{a^x\}\{\backslash ln\; a\}\backslash qquad\backslash text\{\; for\; \}a\; >\; 0,\backslash \; a\; \backslash ne\; 1$

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In the following formulas, {{math|erf}} is the error function and {{math|Ei}} is the exponential integral.

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• $\backslash int\; e^\{cx\}\backslash ln\; x\backslash ,dx\; =\; \backslash frac\{1\}\{c\}\backslash left(e^\{cx\}\backslash ln|x|-\backslash operatorname\{Ei\}(cx)\backslash right)$
• $\backslash int\; x\; e^\{c\; x^2\; \}\backslash ,dx=\; \backslash frac\{1\}\{2c\}\; e^\{c\; x^2\}$
• $\backslash int\; e^\{-c\; x^2\; \}\backslash ,dx=\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{4c\}\}\; \backslash operatorname\{erf\}(\backslash sqrt\{c\}\; x)$
• $\backslash int\; xe^\{-c\; x^2\; \}\backslash ,dx=-\backslash frac\{1\}\{2c\}e^\{-cx^2\}$
• $\backslash int\backslash frac\{e^\{-x^2\}\}\{x^2\}\backslash ,dx\; =\; -\backslash frac\{e^\{-x^2\}\}\{x\}\; -\; \backslash sqrt\{\backslash pi\}\; \backslash operatorname\{erf\}\; (x)$
• $\backslash int\; \{\backslash frac\{1\}\{\backslash sigma\backslash sqrt\{2\backslash pi\}\}\; e^\{\; -\backslash frac\{1\}\{2\}\backslash left(\backslash frac\{x-\backslash mu\}\{\backslash sigma\}\backslash right)^2\; \}\}\backslash ,dx=\; \backslash frac\{1\}\{2\}\backslash operatorname\{erf\}\backslash left(\backslash frac\{x-\backslash mu\}\{\backslash sigma\; \backslash sqrt\{2\}\}\backslash right)$

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• $\backslash int\; e^\{x^2\}\backslash ,dx\; =\; e^\{x^2\}\backslash left(\; \backslash sum\_\{j=0\}^\{n-1\}c\_\{2j\}\backslash frac\{1\}\{x^\{2j+1\}\}\; \backslash right\; )+(2n-1)c\_\{2n-2\}\; \backslash int\; \backslash frac\{e^\{x^2\}\}\{x^\{2n\}\}\backslash ,dx\; \backslash quad\; \backslash text\{valid\; for\; any\; \}\; n\; >\; 0,$

where $c\_\{2j\}=\backslash frac\{\; 1\; \backslash cdot\; 3\; \backslash cdot\; 5\; \backslash cdots\; (2j-1)\}\{2^\{j+1\}\}=\backslash frac\{(2j)!\}\{j!2^\{2j+1\}\}\; \backslash \; .$

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

• $\{\backslash int\; \backslash underbrace\{x^\{x^\{\backslash cdot^\{\backslash cdot^\{x\}\}\}\}\}\_mdx=\; \backslash sum\_\{n=0\}^m\backslash frac\{(-1)^n(n+1)^\{n-1\}\}\{n!\}\backslash Gamma(n+1,-\; \backslash ln\; x)\; +\; \backslash sum\_\{n=m+1\}^\backslash infty(-1)^na\_\{mn\}\backslash Gamma(n+1,-\backslash ln\; x)\; \backslash qquad\backslash text\{(for\; \}x>\; 0\backslash text\{)\}\}$

where

and {{math|Γ(x,y)}} is the upper incomplete gamma function.

• $\backslash int\; \backslash frac\{1\}\{ae^\{\backslash lambda\; x\}\; +\; b\}\; \backslash ,dx\; =\; \backslash frac\{x\}\{b\}\; -\; \backslash frac\{1\}\{b\; \backslash lambda\}\; \backslash ln\backslash left(a\; e^\{\backslash lambda\; x\}\; +\; b\; \backslash right)$ when $b\; \backslash neq\; 0$, $\backslash lambda\; \backslash neq\; 0$, and $ae^\{\backslash lambda\; x\}\; +\; b\; >\; 0.$
• $\backslash int\; \backslash frac\{e^\{2\backslash lambda\; x\}\}\{ae^\{\backslash lambda\; x\}\; +\; b\}\; \backslash ,dx\; =\; \backslash frac\{1\}\{a^2\; \backslash lambda\}\; \backslash left[a\; e^\{\backslash lambda\; x\}\; +\; b\; -\; b\; \backslash ln\backslash left(a\; e^\{\backslash lambda\; x\}\; +\; b\; \backslash right)\; \backslash right]$ when $a\; \backslash neq\; 0$, $\backslash lambda\; \backslash neq\; 0$, and $ae^\{\backslash lambda\; x\}\; +\; b\; >\; 0.$
• $\backslash int\; \backslash frac\{ae^\{cx\}-1\}\{be^\{cx\}-1\}\backslash ,dx=\backslash frac\{(a-b)\backslash log(1-be^\{cx\})\}\{bc\}+x.$
• $\backslash int\{e^\{x\}\backslash left(\; f\backslash left(\; x\; \backslash right)\; +\; f\text{'}\backslash left(\; x\; \backslash right)\; \backslash right)\backslash text\{dx\}\}\; =\; e^\{x\}f\backslash left(\; x\; \backslash right)\; +\; C$

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• $\backslash int\; \{e^\{x\}\backslash left(\; f\backslash left(\; x\; \backslash right)\; -\; \backslash left(\; -\; 1\; \backslash right)^\{n\}\backslash frac\{d^\{n\}f\backslash left(\; x\; \backslash right)\}\{dx^\{n\}\}\; \backslash right)\backslash ,dx\}\; =\; e^\{x\}\backslash sum\_\{k\; =\; 1\}^\{n\}\{\backslash left(\; -\; 1\; \backslash right)^\{k\; -\; 1\}\backslash frac\{d^\{k\; -\; 1\}f\backslash left(\; x\; \backslash right)\}\{dx^\{k\; -\; 1\}\}\}\; +\; C$
• $\backslash int\; \{e^\{-\; x\}\backslash left(\; f\backslash left(\; x\; \backslash right)\; -\; \backslash frac\{d^\{n\}f\backslash left(\; x\; \backslash right)\}\{dx^\{n\}\}\; \backslash right)\backslash ,\; dx\}\; =\; -\; e^\{-\; x\}\backslash sum\_\{k\; =\; 1\}^\{n\}\backslash frac\{d^\{k\; -\; 1\}f\backslash left(\; x\; \backslash right)\}\{dx^\{k\; -\; 1\}\}\; +\; C$

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• $\backslash int\; \{e^\{ax\}\backslash left(\; \backslash left(\; a\backslash right)^\{n\}f\backslash left(\; x\; \backslash right)\; -\; \backslash left(\; -\; 1\; \backslash right)^\{n\}\backslash frac\{d^\{n\}f\backslash left(\; x\; \backslash right)\}\{dx^\{n\}\}\; \backslash right)\backslash ,dx\}\; =\; e^\{ax\}\backslash sum\_\{k\; =\; 1\}^\{n\}\{\backslash left(a\backslash right)^\{n-k\}\backslash left(\; -\; 1\; \backslash right)^\{k\; -\; 1\}\backslash frac\{d^\{k\; -\; 1\}f\backslash left(\; x\; \backslash right)\}\{dx^\{k\; -\; 1\}\}\}\; +\; C$

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• $\backslash begin\{align\}$

\int_0^1 e^{x\cdot \ln a + (1-x)\cdot \ln b}\,dx

&= \int_0^1 \left(\frac{a}{b}\right)^{x}\cdot b\,dx \\

&= \int_0^1 a^{x}\cdot b^{1-x}\,dx \\

&= \frac{a-b}{\ln a - \ln b} \qquad\text{for } a > 0,\ b > 0,\ a \neq b

\end{align}

The last expression is the logarithmic mean.

• $\backslash int\_0^\{\backslash infty\}\; e^\{-ax\}\backslash ,dx=\backslash frac\{1\}\{a\}\; \backslash quad\; (\backslash operatorname\{Re\}(a)>0)$
• $\backslash int\_0^\{\backslash infty\}\; e^\{-ax^2\}\backslash ,dx=\backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash pi\; \backslash over\; a\}\; \backslash quad\; (a>0)$ (the Gaussian integral)
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-ax^2\}\backslash ,dx=\backslash sqrt\{\backslash pi\; \backslash over\; a\}\; \backslash quad\; (a>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-ax^2\}\; e^\{-\backslash frac\{b\}\{x^2\}\}\backslash ,dx=\backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}e^\{-2\backslash sqrt\{ab\}\}\; \backslash quad\; (a,b>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-(ax^2\; +\; bx)\}\backslash ,dx=\; \backslash sqrt\{\backslash pi\; \backslash over\; a\}e^\{\backslash tfrac\{b^2\}\{4a\}\}\; \backslash quad(a\; >\; 0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-(ax^2\; +\; bx+c)\}\backslash ,dx=\; \backslash sqrt\{\backslash pi\; \backslash over\; a\}e^\{\backslash tfrac\{b^2\}\{4a\}-c\}\; \backslash quad(a\; >\; 0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{-ax^2\}\; e^\{-2bx\}\backslash ,dx=\backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}e^\{\backslash frac\{b^2\}\{a\}\}\; \backslash quad\; (a>0)$ (see Integral of a Gaussian function)
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x\; e^\{-a(x-b)^2\}\backslash ,dx=\; b\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{a\}\}\; \backslash quad\; (\backslash operatorname\{Re\}(a)>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x\; e^\{-ax^2+bx\}\backslash ,dx=\; \backslash frac\{\; \backslash sqrt\{\backslash pi\}\; b\; \}\{2a^\{3/2\}\}\; e^\{\backslash frac\{b^2\}\{4a\}\}\; \backslash quad\; (\backslash operatorname\{Re\}(a)>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x^2\; e^\{-ax^2\}\backslash ,dx=\backslash frac\{1\}\{2\}\; \backslash sqrt\{\backslash pi\; \backslash over\; a^3\}\; \backslash quad\; (a>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x^2\; e^\{-(ax^2+bx)\}\backslash ,dx=\backslash frac\{\backslash sqrt\{\backslash pi\}(2a+b^2)\}\{4a^\{5/2\}\}\; e^\{\backslash frac\{b^2\}\{4a\}\}\; \backslash quad\; (\backslash operatorname\{Re\}(a)>0)$
• $\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; x^3\; e^\{-(ax^2+bx)\}\backslash ,dx=\backslash frac\{\backslash sqrt\{\backslash pi\}(6a+b^2)b\}\{8a^\{7/2\}\}\; e^\{\backslash frac\{b^2\}\{4a\}\}\; \backslash quad\; (\backslash operatorname\{Re\}(a)>0)$
• $\backslash int\_0^\{\backslash infty\}\; x^\{n\}\; e^\{-ax^2\}\backslash ,dx\; =$

\begin{cases}

\dfrac{\Gamma \left(\frac{n+1}{2}\right)}{2 a^\frac{n+1}{2} } & (n>-1,\ a>0) \\

\dfrac{(2k-1)!!}{2^{k+1}a^k}\sqrt{\dfrac{\pi}{a}} & (n=2k,\ k \text{ integer},\ a>0) \\

\dfrac{k!}{2(a^{k+1})} & (n=2k+1,\ k \text{ integer},\ a>0)

\end{cases}

(the operator $!!$ is the Double factorial)

• $\backslash int\_0^\{\backslash infty\}\; x^n\; e^\{-ax\}\backslash ,dx\; =$

\begin{cases}

\dfrac{\Gamma(n+1)}{a^{n+1}} & (n>-1,\ \operatorname{Re}(a)>0) \\ \\

\dfrac{n!}{a^{n+1}} & (n=0,1,2,\ldots,\ \operatorname{Re}(a)>0)

\end{cases}

• $\backslash int\_0^\{1\}\; x^n\; e^\{-ax\}\backslash ,dx\; =$

\frac{n!}{a^{n+1}}\left[1-e^{-a}\sum_{i=0}^{n} \frac{a^i}{i!}\right]

• $\backslash int\_0^\{b\}\; x^n\; e^\{-ax\}\backslash ,dx\; =$

\frac{n!}{a^{n+1}}\left[1-e^{-ab}\sum_{i=0}^{n} \frac{(ab)^i}{i!}\right]

• $\backslash int\_0^\backslash infty\; e^\{-ax^b\}\; dx\; =\; \backslash frac\{1\}\{b\}\backslash \; a^\{-\backslash frac\{1\}\{b\}\}\backslash Gamma\backslash left(\backslash frac\{1\}\{b\}\backslash right)$
• $\backslash int\_0^\backslash infty\; x^n\; e^\{-ax^b\}\; dx\; =\; \backslash frac\{1\}\{b\}\backslash \; a^\{-\backslash frac\{n+1\}\{b\}\}\backslash Gamma\backslash left(\backslash frac\{n+1\}\{b\}\backslash right)$
• $\backslash int\_0^\{\backslash infty\}\; e^\{-ax\}\backslash sin\; bx\backslash ,dx\; =\; \backslash frac\{b\}\{a^2+b^2\}\; \backslash quad\; (a>0)$
• $\backslash int\_0^\{\backslash infty\}\; e^\{-ax\}\backslash cos\; bx\backslash ,dx\; =\; \backslash frac\{a\}\{a^2+b^2\}\; \backslash quad\; (a>0)$
• $\backslash int\_0^\{\backslash infty\}\; xe^\{-ax\}\backslash sin\; bx\backslash ,dx\; =\; \backslash frac\{2ab\}\{(a^2+b^2)^2\}\; \backslash quad\; (a>0)$
• $\backslash int\_0^\{\backslash infty\}\; xe^\{-ax\}\backslash cos\; bx\backslash ,dx\; =\; \backslash frac\{a^2-b^2\}\{(a^2+b^2)^2\}\; \backslash quad\; (a>0)$
• $\backslash int\_0^\{\backslash infty\}\; \backslash frac\{e^\{-ax\}\backslash sin\; bx\}\{x\}\backslash ,dx=\backslash arctan\; \backslash frac\{b\}\{a\}$
• $\backslash int\_0^\{\backslash infty\}\; \backslash frac\{e^\{-ax\}-e^\{-bx\}\}\{x\}\backslash ,dx=\backslash ln\; \backslash frac\{b\}\{a\}$
• $\backslash int\_0^\{\backslash infty\}\; \backslash frac\{e^\{-ax\}-e^\{-bx\}\}\{x\}\; \backslash sin\; px\; \backslash ,\; dx=\backslash arctan\; \backslash frac\{b\}\{p\}\; -\; \backslash arctan\; \backslash frac\{a\}\{p\}$
• $\backslash int\_0^\{\backslash infty\}\; \backslash frac\{e^\{-ax\}-e^\{-bx\}\}\{x\}\; \backslash cos\; px\; \backslash ,\; dx=\backslash frac\{1\}\{2\}\; \backslash ln\; \backslash frac\{b^2+p^2\}\{a^2+p^2\}$
• $\backslash int\_0^\{\backslash infty\}\; \backslash frac\{e^\{-ax\}\; (1-\backslash cos\; x)\}\{x^2\}\backslash ,dx=\backslash arccot\; a\; -\; \backslash frac\{a\}\{2\}\backslash ln\; \backslash Big(\backslash frac\{1\}\{a^2\}+1\backslash Big)$
• $\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{a\; x^4+b\; x^3+c\; x^2+d\; x+f\}\; \backslash ,\; dx$

= e^f \sum_{n,m,p=0}^\infty \frac{ b^{4n}}{(4n)!} \frac{c^{2m}}{(2m)!} \frac{d^{4p}}{(4p)!} \frac{ \Gamma(3n+m+p+\frac14) }{a^{3n+m+p+\frac14} } (appears in several models of extended superstring theory in higher dimensions)

• $\backslash int\_0^\{2\; \backslash pi\}\; e^\{x\; \backslash cos\; \backslash theta\}\; d\; \backslash theta\; =\; 2\; \backslash pi\; I\_0(x)$ ({{math|I₀}} is the modified Bessel function of the first kind)
• $\backslash int\_0^\{2\; \backslash pi\}\; e^\{x\; \backslash cos\; \backslash theta\; +\; y\; \backslash sin\; \backslash theta\}\; d\; \backslash theta\; =\; 2\; \backslash pi\; I\_0\; \backslash left(\; \backslash sqrt\{x^2\; +\; y^2\}\; \backslash right)$
• $\backslash int\_0^\backslash infty\backslash frac\{x^\{s-1\}\}\{e^x/z-1\}\; \backslash ,dx\; =\; \backslash operatorname\{Li\}\_\{s\}(z)\backslash Gamma(s),$

where $\backslash operatorname\{Li\}\_\{s\}(z)$ is the Polylogarithm.

• $\backslash int\_0^\backslash infty\backslash frac\{\backslash sin\; mx\}\{e^\{2\; \backslash pi\; x\}-1\}\; \backslash ,dx\; =\; \backslash frac\{1\}\{4\}\; \backslash coth\; \backslash frac\{m\}\{2\}\; -\; \backslash frac\{1\}\{2m\}$
• $\backslash int\_0^\backslash infty\; e^\{-x\}\; \backslash ln\; x\backslash ,\; dx\; =\; -\; \backslash gamma,$

where $\backslash gamma$ is the Euler–Mascheroni constant which equals the value of a number of definite integrals.

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Finally, a well known result,

$$\backslash int\_0^\{2\; \backslash pi\}\; e^\{i(m-n)\backslash phi\}\; d\backslash phi\; =\; 2\; \backslash pi\; \backslash delta\_\{m,n\}\; \backslash qquad\backslash text\{for\; \}m,n\backslash in\backslash mathbb\{Z\}$$

where $\backslash delta\_\{m,n\}$ is the Kronecker delta.

• Gradshteyn and Ryzhik

{{Reflist|30em}}

Toyesh Prakash Sharma, Etisha Sharma, "Putting Forward Another Generalization Of The Class Of Exponential Integrals And Their Applications.," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.2, pp.1-8, 2023.[https://www.isroset.org/pdf_paper_view.php?paper_id=3100&1-ISROSET-IJSRMSS-08692.pdf]

• {{cite book |first=Victor Hugo |last=Moll |author-link=Victor Hugo Moll |title=Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume I |volume=I |edition=1 |work=Series: Monographs and Research Notes in Mathematics |publisher=Chapman and Hall/CRC Press |date=2014-11-12 |isbn=978-1-48225-651-2 |url=http://www.crcpress.com/Special-Integrals-of-Gradshteyn-and-Ryzhik-the-Proofs---Volume-I/Moll/9781482256512 |access-date=2016-02-12}}
• {{cite book |first=Victor Hugo |last=Moll |author-link=Victor Hugo Moll |title=Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume II |volume=II |edition=1 |work=Series: Monographs and Research Notes in Mathematics |publisher=Chapman and Hall/CRC Press |date=2015-10-27 |isbn=978-1-48225-653-6 |url=http://www.crcpress.com/Special-Integrals-of-Gradshteyn-and-Ryzhik-the-Proofs---Volume-II/Moll/9781482256536 |access-date=2016-02-12}}
• Toyesh Prakash Sharma, https://www.isroset.org/pdf_paper_view.php?paper_id=2214&7-ISROSET-IJSRMSS-05130.pdf

• [http://www.wolframalpha.com/calculators/integral-calculator/ Wolfram Mathematica Online Integrator]
• {{cite web |first=Victor Hugo |last=Moll |author-link=Victor Hugo Moll |title=List with the formulas and proofs in GR |url=http://www.math.tulane.edu/~vhm/Table.html |access-date=2016-02-12}}

{{Lists of integrals}}

{{DEFAULTSORT:Integrals Of Exponential Functions}}

Category:Exponentials

Exponential functions