List of integrals of Gaussian functions


In these expressions,

is the standard normal probability density function,

is the corresponding cumulative distribution function (where erf is the error function) and

is Owen's T function.

Owen[nb 1] has an extensive list of Gaussian-type integrals; only a subset is given below.

Indefinite integrals

[nb 2]

In these integrals, n!! is the double factorial: for even n it is equal to the product of all even numbers from 2 to n, and for odd n it is the product of all odd numbers from 1 to n ; additionally it is assumed that 0!! = (−1)!! = 1.

[nb 3]

Definite integrals

[nb 4]


  1. ^ Owen (1980)
  2. ^ Patel & Read (1996) lists this integral above without the minus sign, which is an error. See calculation by WolframAlpha
  3. ^ Patel & Read (1996) report this integral with error, see WolframAlpha
  4. ^ Patel & Read (1996) report this integral incorrectly by omitting x from the integrand
  • Patel, Jagdish K.; Read, Campbell B. (1996). Handbook of the normal distribution (2nd ed.). CRC Press. ISBN 0-8247-9342-0.
  • Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation. B9 (4): 389–419. doi:10.1080/03610918008812164.