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.
Integrals of polynomials
[edit]Integrals involving only exponential functions
[edit]Integrals involving the error function
[edit]In the following formulas, erf is the error function and Ei is the exponential integral.
Other integrals
[edit]where
(Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.)
where
and Γ(x,y) is the upper incomplete gamma function.
- when , , and
- when , , and
Definite integrals
[edit]The last expression is the logarithmic mean.
- (the Gaussian integral)
- (see Integral of a Gaussian function)
(the operator is the Double factorial)
- (appears in several models of extended superstring theory in higher dimensions)
- (I0 is the modified Bessel function of the first kind)
where is the Polylogarithm.
where is the Euler–Mascheroni constant which equals the value of a number of definite integrals.
Finally, a well known result, where is the Kronecker delta.
See also
[edit]References
[edit]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
[edit]- Moll, Victor Hugo (2014-11-12). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume I. Vol. I (1 ed.). Chapman and Hall/CRC Press. ISBN 978-1-48225-651-2. Retrieved 2016-02-12.
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ignored (help) - Moll, Victor Hugo (2015-10-27). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume II. Vol. II (1 ed.). Chapman and Hall/CRC Press. ISBN 978-1-48225-653-6. Retrieved 2016-02-12.
{{cite book}}
:|work=
ignored (help) - Toyesh Prakash Sharma, https://www.isroset.org/pdf_paper_view.php?paper_id=2214&7-ISROSET-IJSRMSS-05130.pdf
External links
[edit]- Wolfram Mathematica Online Integrator
- Moll, Victor Hugo. "List with the formulas and proofs in GR". Retrieved 2016-02-12.