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

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

Integrals involving only exponential functions

Integrals involving the error function

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

Other integrals

(Note that the value of the expression is independent of the value of n, which is why it does not appear in the integral.)
and Γ(x,y) is the upper incomplete gamma function.
when , , and
when , , and

the below formulae was proved by Toyesh Prakash Sharma.[citation needed]

(if is a positive integer)
(if is a positive integer)

Definite integrals

The last expression is the logarithmic mean.

(the Gaussian integral)

(see Integral of a Gaussian function)

(the operator is the Double factorial)

(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,

(For integer m, n)

where is the Kronecker delta.

See also


Further reading