List of integrals of Gaussian functions

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In these expressions

is the standard normal probability density function,

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

is known as the Owen's T function.

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

Indefinite integrals[edit]

[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[edit]

[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. pp. 389–419.