List of integrals of exponential functions

Short description: 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.

## Contents

* 1 Indefinite integral
* 1.1 Integrals of polynomials
* 1.2 Integrals involving only exponential functions
* 1.3 Integrals involving the error function
* 1.4 Other integrals
* 2 Definite integrals
* 3 See also
* 4 References
* 5 Further reading
* 6 External links

## 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 of polynomials

* ∫xecxdx=ecx(cx−1c2) for c≠0;
* ∫x2ecxdx=ecx(x2c−2xc2+2c3)
* ∫xnecxdx=1cxnecx−nc∫xn−1ecxdx=(∂∂c)necxc=ecx∑i=0n(−1)in!(n−i)!ci+1xn−i=ecx∑i=0n(−1)n−in!i!cn−i+1xi
* ∫ecxxdx=ln⁡|x|+∑n=1∞(cx)nn⋅n!
* ∫ecxxndx=1n−1(−ecxxn−1+c∫ecxxn−1dx)(for n≠1)

### Integrals involving only exponential functions

* ∫f′(x)ef(x)dx=ef(x)
* ∫ecxdx=1cecx
* ∫axdx=axln⁡a for a>0, a≠1

### Integrals involving the error function

In the following formulas, erf is the error function and Ei is the exponential integral.

* ∫ecxln⁡xdx=1c(ecxln⁡|x|−Ei⁡(cx))
* ∫xecx2dx=12cecx2
* ∫e−cx2dx=π4cerf⁡(cx)
* ∫xe−cx2dx=−12ce−cx2
* ∫e−x2x2dx=−e−x2x−πerf⁡(x)
* ∫1σ2πe−12(x−μσ)2dx=12erf⁡(x−μσ2)

### Other integrals

* ∫ex2dx=ex2(∑j=0n−1c2j1x2j+1)+(2n−1)c2n−2∫ex2x2ndxvalid for any n>0,

where c2j=1⋅3⋅5⋯(2j−1)2j+1=(2j)!j!22j+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.)

* ∫xx⋅⋅x⏟mdx=∑n=0m(−1)n(n+1)n−1n!Γ(n+1,−ln⁡x)+∑n=m+1∞(−1)namnΓ(n+1,−ln⁡x)(for x>0)

where amn={1if n=0,1n!if m=1,1n∑j=1njam,n−jam−1,j−1otherwise

and Γ(x,y) is the upper incomplete gamma function.

* ∫1aeλx+bdx=xb−1bλln⁡(aeλx+b) when b≠0, λ≠0, and aeλx+b>0.
* ∫e2λxaeλx+bdx=1a2λ[aeλx+b−bln⁡(aeλx+b)] when a≠0, λ≠0, and aeλx+b>0.
* ∫aecx−1becx−1dx=(a−b)log⁡(1−becx)bc+x.
* ∫ex(f(x)+f′(x))dx=exf(x)+C

* ∫ex(f(x)−(−1)ndnf(x)dxn)dx=ex∑k=1n(−1)k−1dk−1f(x)dxk−1+C
* ∫e−x(f(x)−dnf(x)dxn)dx=−e−x∑k=1ndk−1f(x)dxk−1+C

* ∫eax((a)nf(x)−(−1)ndnf(x)dxn)dx=eax∑k=1n(a)n−k(−1)k−1dk−1f(x)dxk−1+C

## Definite integrals

* ∫01ex⋅ln⁡a+(1−x)⋅ln⁡bdx=∫01(ab)x⋅bdx=∫01ax⋅b1−xdx=a−bln⁡a−ln⁡bfor a>0, b>0, a≠b

The last expression is the logarithmic mean.

* ∫0∞e−axdx=1a(Re⁡(a)>0)
* ∫0∞e−ax2dx=12πa(a>0) (the Gaussian integral)
* ∫−∞∞e−ax2dx=πa(a>0)
* ∫−∞∞e−ax2e−bx2dx=πae−2ab(a,b>0)
* ∫−∞∞e−(ax2+bx)dx=πaeb24a(a>0)
* ∫−∞∞e−(ax2+bx+c)dx=πaeb24a−c(a>0)
* ∫−∞∞e−ax2e−2bxdx=πaeb2a(a>0) (see Integral of a Gaussian function)
* ∫−∞∞xe−a(x−b)2dx=bπa(Re⁡(a)>0)
* ∫−∞∞xe−ax2+bxdx=πb2a3/2eb24a(Re⁡(a)>0)
* ∫−∞∞x2e−ax2dx=12πa3(a>0)
* ∫−∞∞x2e−(ax2+bx)dx=π(2a+b2)4a5/2eb24a(Re⁡(a)>0)
* ∫−∞∞x3e−(ax2+bx)dx=π(6a+b2)b8a7/2eb24a(Re⁡(a)>0)
* ∫0∞xne−ax2dx={Γ(n+12)2(an+12)(n>−1, a>0)(2k−1)!!2k+1akπa(n=2k, k integer, a>0)k!2(ak+1)(n=2k+1, k integer, a>0)

(the operator !! is the Double factorial)

* ∫0∞xne−axdx={Γ(n+1)an+1(n>−1, Re⁡(a)>0)n!an+1(n=0,1,2,…, Re⁡(a)>0)
* ∫01xne−axdx=n!an+1[1−e−a∑i=0naii!]
* ∫0bxne−axdx=n!an+1[1−e−ab∑i=0n(ab)ii!]
* ∫0∞e−axbdx=1b a−1bΓ(1b)
* ∫0∞xne−axbdx=1b a−n+1bΓ(n+1b)
* ∫0∞e−axsin⁡bxdx=ba2+b2(a>0)
* ∫0∞e−axcos⁡bxdx=aa2+b2(a>0)
* ∫0∞xe−axsin⁡bxdx=2ab(a2+b2)2(a>0)
* ∫0∞xe−axcos⁡bxdx=a2−b2(a2+b2)2(a>0)
* ∫0∞e−axsin⁡bxxdx=arctan⁡ba
* ∫0∞e−ax−e−bxxdx=ln⁡ba
* ∫0∞e−ax−e−bxxsin⁡pxdx=arctan⁡bp−arctan⁡ap
* ∫0∞e−ax−e−bxxcos⁡pxdx=12ln⁡b2+p2a2+p2
* ∫0∞e−ax(1−cos⁡x)x2dx=arccot⁡a−a2ln⁡(1a2+1)
* ∫−∞∞eax4+bx3+cx2+dx+fdx=ef∑n,m,p=0∞b4n(4n)!c2m(2m)!d4p(4p)!Γ(3n+m+p+14)a3n+m+p+14 (appears in several models of extended superstring theory in higher dimensions)
* ∫02πexcos⁡θdθ=2πI0(x) (I0 is the modified Bessel function of the first kind)
* ∫02πexcos⁡θ+ysin⁡θdθ=2πI0(x2+y2)
* ∫0∞xs−1ex/z−1dx=Lis(z)Γ(s),

where Lis(z) is the Polylogarithm.

* ∫0∞sin⁡mxe2πx−1dx=14coth⁡m2−12m
* ∫0∞e−xln⁡xdx=−γ,

where γ is the Euler–Mascheroni constant which equals the value of a number of definite integrals.

Finally, a well known result, ∫02πei(m−n)ϕdϕ=2πδm,nfor m,n∈ℤ where δm,n is the Kronecker delta.

## See also

* Gradshteyn and Ryzhik

## References

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.[1]

## Further reading

* Moll, Victor Hugo (2014-11-12). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume I. I (1 ed.). Chapman and Hall/CRC Press. ISBN 978-1-48225-651-2. http://www.crcpress.com/Special-Integrals-of-Gradshteyn-and-Ryzhik-the-Proofs
Volume-I/Moll/9781482256512. Retrieved 2016-02-12.
* Moll, Victor Hugo (2015-10-27). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume II. II (1 ed.). Chapman and Hall/CRC Press. ISBN 978-1-48225-653-6. http://www.crcpress.com/Special-Integrals-of-Gradshteyn-and-Ryzhik-the-Proofs
Volume-II/Moll/9781482256536. Retrieved 2016-02-12.
* Toyesh Prakash Sharma, https://www.isroset.org/pdf_paper_view.php?paper_id=2214&7-ISROSET-IJSRMSS-05130.pdf

## External links

* Wolfram Mathematica Online Integrator
* Moll, Victor Hugo. "List with the formulas and proofs in GR". http://www.math.tulane.edu/~vhm/Table.html.

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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