View source for List of integrals of exponential functions

{{short description|List Of Integrals Of Exponential Functions}}
The following is a list of [[Integral|integral]]s of [[Exponential function|exponential function]]s. For a complete list of integral functions, please see the list of integrals.

== Indefinite integral ==

Indefinite integrals are [[Antiderivative|antiderivative]] functions. A constant (the [[Constant of integration|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 of polynomials===

{{startplainlist|indent=1}}
* <math>
\int xe^{cx}\,dx = e^{cx}\left(\frac{cx-1}{c^{2}}\right) \qquad \text{ for } c \neq 0;
</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>\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}</math>
* <math>\int\frac{e^{cx}}{x}\,dx = \ln|x| +\sum_{n=1}^\infty\frac{(cx)^n}{n\cdot n!}</math>
* <math>\int\frac{e^{cx}}{x^n}\,dx = \frac{1}{n-1}\left(-\frac{e^{cx}}{x^{n-1}}+c\int\frac{e^{cx} }{x^{n-1}}\,dx\right) \qquad\text{(for }n\neq 1\text{)}</math>

===Integrals involving only exponential functions===

{{startplainlist|indent=1}}
* <math>\int  f'(x)e^{f(x)}\,dx = e^{f(x)}</math>
* <math>\int e^{cx}\,dx = \frac{1}{c} e^{cx}</math>
* <math>\int a^{x}\,dx = \frac{a^x}{\ln a}\qquad\text{ for }a > 0,\ a \ne 1</math>

===Integrals involving the error function===

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

{{startplainlist|indent=1}}
* <math>\int e^{cx}\ln x\,dx = \frac{1}{c}\left(e^{cx}\ln|x|-\operatorname{Ei}(cx)\right)</math>
* <math>\int x e^{c x^2 }\,dx= \frac{1}{2c} e^{c x^2}</math>
* <math>\int e^{-c x^2 }\,dx= \sqrt{\frac{\pi}{4c}} \operatorname{erf}(\sqrt{c} x)</math>
* <math>\int xe^{-c x^2 }\,dx=-\frac{1}{2c}e^{-cx^2} </math>
* <math>\int\frac{e^{-x^2}}{x^2}\,dx = -\frac{e^{-x^2}}{x} - \sqrt{\pi} \operatorname{erf} (x) </math>
* <math>\int {\frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }}\,dx= \frac{1}{2}\operatorname{erf}\left(\frac{x-\mu}{\sigma \sqrt{2}}\right)</math>

===Other integrals===

{{startplainlist|indent=1}}
* <math>\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 \text{valid for any } n > 0,
  </math><p>where <math> c_{2j}=\frac{ 1 \cdot 3 \cdot 5 \cdots (2j-1)}{2^{j+1}}=\frac{(2j)!}{j!2^{2j+1}} \ . </math></p>
  <p>(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.)</p>
* <math> {\int \underbrace{x^{x^{\cdot^{\cdot^{x}}}}}_mdx= \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\text{(for }x> 0\text{)}}</math>
  <p>where <math>a_{mn}=\begin{cases}1   &\text{if } n = 0, \\ \\ \dfrac{1}{n!} &\text{if } m=1, \\ \\ \dfrac{1}{n}\sum_{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}  &\text{otherwise} \end{cases}</math></p>
  <p>and {{math|Γ(''x'',''y'')}} is the [[Incomplete gamma function#Definition|upper incomplete gamma function]].</p>
* <math>\int \frac{1}{ae^{\lambda x} + b} \,dx = \frac{x}{b} - \frac{1}{b \lambda} \ln\left(a e^{\lambda x} + b \right) </math> when <math>b \neq 0</math>, <math>\lambda \neq 0</math>, and <math>ae^{\lambda x} + b > 0.</math>
* <math>\int \frac{e^{2\lambda x}}{ae^{\lambda x} + b} \,dx = \frac{1}{a^2 \lambda} \left[a e^{\lambda x} + b - b \ln\left(a e^{\lambda x} + b \right) \right] </math>  when <math>a \neq 0</math>, <math>\lambda \neq 0</math>, and <math>ae^{\lambda x} + b > 0.</math>
* <math>\int \frac{ae^{cx}-1}{be^{cx}-1}\,dx=\frac{(a-b)\log(1-be^{cx})}{bc}+x.</math>
* <math>\int{e^{x}\left( f\left( x \right) + f'\left( x \right) \right)\text{dx}} = e^{x}f\left( x \right) + C</math>

{{startplainlist|indent=1}}
* <math>\int {e^{x}\left( f\left( x \right) - \left( - 1 \right)^{n}\frac{d^{n}f\left( x \right)}{dx^{n}} \right)\,dx} = e^{x}\sum_{k = 1}^{n}{\left( - 1 \right)^{k - 1}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}}} + C</math>
* <math>\int {e^{- x}\left( f\left( x \right) - \frac{d^{n}f\left( x \right)}{dx^{n}} \right)\, dx} = - e^{- x}\sum_{k = 1}^{n}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}} + C</math>

* <math>\int {e^{ax}\left( \left( a\right)^{n}f\left( x \right) - \left( - 1 \right)^{n}\frac{d^{n}f\left( x \right)}{dx^{n}} \right)\,dx} = e^{ax}\sum_{k = 1}^{n}{\left(a\right)^{n-k}\left( - 1 \right)^{k - 1}\frac{d^{k - 1}f\left( x \right)}{dx^{k - 1}}} + C</math>

== Definite integrals ==

{{startplainlist|indent=1}}
* <math>\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}</math>
<p>The last expression is the [[Logarithmic mean|logarithmic mean]].</p>
* <math>\int_0^{\infty} e^{-ax}\,dx=\frac{1}{a} \quad (\operatorname{Re}(a)>0)</math>
* <math>\int_0^{\infty} e^{-ax^2}\,dx=\frac{1}{2} \sqrt{\pi \over a} \quad  (a>0)</math> (the [[Gaussian integral]])
* <math>\int_{-\infty}^{\infty} e^{-ax^2}\,dx=\sqrt{\pi \over a} \quad (a>0)</math>
* <math>\int_{-\infty}^{\infty} e^{-ax^2} e^{-\frac{b}{x^2}}\,dx=\sqrt{\frac{\pi}{a}}e^{-2\sqrt{ab}} \quad (a,b>0) </math>
* <math>\int_{-\infty}^{\infty} e^{-(ax^2 + bx)}\,dx= \sqrt{\pi \over a}e^{\tfrac{b^2}{4a}} \quad(a > 0)</math>
* <math>\int_{-\infty}^{\infty} e^{-(ax^2 + bx+c)}\,dx= \sqrt{\pi \over a}e^{\tfrac{b^2}{4a}-c} \quad(a > 0)</math>
* <math>\int_{-\infty}^{\infty} e^{-ax^2} e^{-2bx}\,dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{a}} \quad (a>0)</math> (see Integral of a Gaussian function)
* <math>\int_{-\infty}^{\infty} x e^{-a(x-b)^2}\,dx= b \sqrt{\frac{\pi}{a}} \quad (\operatorname{Re}(a)>0)</math>
* <math>\int_{-\infty}^{\infty} x e^{-ax^2+bx}\,dx= \frac{ \sqrt{\pi} b }{2a^{3/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)</math>
* <math>\int_{-\infty}^{\infty} x^2 e^{-ax^2}\,dx=\frac{1}{2} \sqrt{\pi \over a^3} \quad (a>0)</math>
* <math>\int_{-\infty}^{\infty} x^2 e^{-(ax^2+bx)}\,dx=\frac{\sqrt{\pi}(2a+b^2)}{4a^{5/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)</math>
* <math>\int_{-\infty}^{\infty} x^3 e^{-(ax^2+bx)}\,dx=\frac{\sqrt{\pi}(6a+b^2)b}{8a^{7/2}} e^{\frac{b^2}{4a}} \quad (\operatorname{Re}(a)>0)</math>
* <math>\int_0^{\infty} x^{n} e^{-ax^2}\,dx = 
  \begin{cases}
       \dfrac{\Gamma \left(\frac{n+1}{2}\right)}{2\left(a^\frac{n+1}{2}\right) } & (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}</math>
  <p>(the operator <math>!!</math> is the [[Double factorial]])</p>
* <math>\int_0^{\infty} x^n e^{-ax}\,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}</math>
* <math>\int_0^{1} x^n e^{-ax}\,dx = 
  \frac{n!}{a^{n+1}}\left[1-e^{-a}\sum_{i=0}^{n} \frac{a^i}{i!}\right]</math>
* <math>\int_0^{b} x^n e^{-ax}\,dx =
  \frac{n!}{a^{n+1}}\left[1-e^{-ab}\sum_{i=0}^{n} \frac{(ab)^i}{i!}\right]</math>
* <math>\int_0^\infty e^{-ax^b} dx =  \frac{1}{b}\ a^{-\frac{1}{b}}\Gamma\left(\frac{1}{b}\right)</math>
* <math>\int_0^\infty x^n e^{-ax^b} dx = \frac{1}{b}\ a^{-\frac{n+1}{b}}\Gamma\left(\frac{n+1}{b}\right)</math>
* <math>\int_0^{\infty} e^{-ax}\sin bx\,dx = \frac{b}{a^2+b^2} \quad (a>0)</math>
* <math>\int_0^{\infty} e^{-ax}\cos bx\,dx = \frac{a}{a^2+b^2} \quad (a>0)</math>
* <math>\int_0^{\infty} xe^{-ax}\sin bx\,dx = \frac{2ab}{(a^2+b^2)^2} \quad (a>0)</math>
* <math>\int_0^{\infty} xe^{-ax}\cos bx\,dx = \frac{a^2-b^2}{(a^2+b^2)^2} \quad (a>0)</math>
* <math>\int_0^{\infty} \frac{e^{-ax}\sin bx}{x}\,dx=\arctan \frac{b}{a}</math>
* <math>\int_0^{\infty} \frac{e^{-ax}-e^{-bx}}{x}\,dx=\ln \frac{b}{a}</math>
* <math>\int_0^{\infty} \frac{e^{-ax}-e^{-bx}}{x} \sin px \, dx=\arctan \frac{b}{p} - \arctan \frac{a}{p}</math>
* <math>\int_0^{\infty} \frac{e^{-ax}-e^{-bx}}{x} \cos px \, dx=\frac{1}{2} \ln \frac{b^2+p^2}{a^2+p^2}</math>
* <math>\int_0^{\infty} \frac{e^{-ax} (1-\cos x)}{x^2}\,dx=\arccot a - \frac{a}{2}\ln \Big(\frac{1}{a^2}+1\Big)</math> 
* <math> \int_{-\infty}^\infty e^{a x^4+b x^3+c x^2+d x+f} \, 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} } </math> (appears in several models of extended [[Physics:Superstring theory|superstring theory]] in higher dimensions)
* <math>\int_0^{2 \pi} e^{x \cos \theta} d \theta = 2 \pi I_0(x)</math> ({{math|''I''<sub>0</sub>}} is the [[Bessel function#Modified Bessel functions: Iα, Kα|modified Bessel function]] of the first kind)
* <math>\int_0^{2 \pi} e^{x \cos \theta + y \sin \theta} d \theta = 2 \pi I_0 \left( \sqrt{x^2 + y^2} \right)</math>
* <math>\int_0^\infty\frac{x^{s-1}}{e^x/z-1} \,dx = \operatorname{Li}_{s}(z)\Gamma(s), </math>
  <p>where <math>\operatorname{Li}_{s}(z)</math> is the [[Polylogarithm#Integral representations|Polylogarithm]].</p>
* <math>\int_0^\infty\frac{\sin mx}{e^{2 \pi x}-1} \,dx = \frac{1}{4} \coth \frac{m}{2} - \frac{1}{2m} </math>
* <math>\int_0^\infty  e^{-x} \ln x\, dx = - \gamma, </math>
  <p>where <math>\gamma</math> is the [[Euler–Mascheroni constant#Integrals|Euler–Mascheroni constant]] which equals the value of a number of definite integrals.</p>

Finally, a well known result,
<math display="block">\int_0^{2 \pi} e^{i(m-n)\phi} d\phi = 2 \pi \delta_{m,n} \qquad\text{for }m,n\in\mathbb{Z}</math>
where <math>\delta_{m,n}</math> is the [[Kronecker delta]].

==See also==
*[[Gradshteyn and Ryzhik]]

== References ==
{{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]

==Further reading==
*{{cite book |first=Victor Hugo |last=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 |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

==External links==
* [http://www.wolframalpha.com/calculators/integral-calculator/ Wolfram Mathematica Online Integrator]
* {{cite web |first=Victor Hugo |last=Moll |title=<!-- This is a -->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]]
[[Category:Lists of integrals|Exponential functions]]

{{Sourceattribution|List of integrals of exponential functions}}