List of integrals of Gaussian functions

In these expressions

${\displaystyle \phi (x)={\frac {e^{-{\frac {x^{2}}{2}}}}{\sqrt {2\pi }}}}$

is the standard normal probability density function,

${\displaystyle \Phi (x)=\int _{-\infty }^{x}\phi (t)dt={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}$

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

${\displaystyle T(h,a)=\phi (h)\int _{0}^{a}{\frac {\phi (hx)}{1+x^{2}}}\,dx}$

which 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

${\displaystyle \int \phi (x)\,dx=\Phi (x)+C}$ 

${\displaystyle \int x\phi (x)\,dx=-\phi (x)+C}$ 

${\displaystyle \int x^{2}\phi (x)\,dx=\Phi (x)-x\phi (x)+C}$ 

${\displaystyle \int x^{2k+1}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k)!!}{(2j)!!}}x^{2j}+C}$ ^{[nb 2]}

${\displaystyle \int x^{2k+2}\phi (x)\,dx=-\phi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{2j+1}+(2k+1)!!\,\Phi (x)+C}$ 

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

${\displaystyle \int \phi (x)^{2}\,dx={\tfrac {1}{2{\sqrt {\pi }}}}\Phi (x{\sqrt {2}})+C}$ 

${\displaystyle \int \phi (x)\phi (a+bx)\,dx={\tfrac {1}{t}}\phi ({\tfrac {a}{t}})\Phi (tx+{\tfrac {ab}{t}})+C,\qquad t={\sqrt {1+b^{2}}}}$ ^{[nb 3]}

${\displaystyle \int x\phi (a+bx)\,dx=-b^{-2}\left(\phi (a+bx)-a\Phi (a+bx)\right)+C}$ 

${\displaystyle \int x^{2}\phi (a+bx)\,dx=b^{-3}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\phi (a+bx)\right)+C}$ 

${\displaystyle \int \phi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C}$ 

${\displaystyle \int \Phi (a+bx)\,dx=b^{-1}\left((a+bx)\Phi (a+bx)+\phi (a+bx)\right)+C}$ 

${\displaystyle \int x\Phi (a+bx)\,dx={\tfrac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\phi (a+bx)\right)+C}$ 

${\displaystyle \int x^{2}\Phi (a+bx)\,dx={\tfrac {1}{3b^{3}}}\left((b^{3}x^{3}+a^{3}+3a)\Phi (a+bx)+(b^{2}x^{2}-abx+a^{2}+2)\phi (a+bx)\right)+C}$ 

${\displaystyle \int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{n+1}-nx^{n-1}\right)\Phi (x)+x^{n}\phi (x)+n(n-1)\int x^{n-2}\Phi (x)\,dx\right)+C}$ 

${\displaystyle \int x\phi (x)\Phi (a+bx)\,dx={\tfrac {b}{t}}\phi ({\tfrac {a}{t}})\Phi (xt+{\tfrac {ab}{t}})-\phi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}}$ 

${\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\phi (x)-{\tfrac {1}{\sqrt {\pi }}}\Phi (x{\sqrt {2}})+C}$ 

${\displaystyle \int e^{cx}\phi (bx)^{n}\,dx={\frac {e^{\frac {c^{2}}{2nb^{2}}}}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\frac {b^{2}xn-c}{b{\sqrt {n}}}}\right)+C,\qquad b\neq 0,n>0}$ 

Definite integrals

${\displaystyle \int _{-\infty }^{\infty }x^{2}\phi (x)^{n}\,dx={\frac {1}{\sqrt {n^{3}(2\pi )^{n-1}}}}}$ 

${\displaystyle \int _{-\infty }^{0}\phi (ax)\Phi (bx)dx=(2\pi |a|)^{-1}\left({\tfrac {\pi }{2}}-\arctan({\tfrac {b}{|a|}})\right)}$ 

${\displaystyle \int _{0}^{\infty }\phi (ax)\Phi (bx)\,dx=(2\pi |a|)^{-1}\left({\tfrac {\pi }{2}}+\arctan({\tfrac {b}{|a|}})\right)}$ 

${\displaystyle \int _{0}^{\infty }x\phi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{\sqrt {1+b^{2}}}}\right)}$ 

${\displaystyle \int _{0}^{\infty }x^{2}\phi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right)}$ 

${\displaystyle \int x\phi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}}$ 

${\displaystyle \int _{0}^{\infty }\Phi (bx)^{2}\phi (x)\,dx=(2\pi )^{-1}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)}$ 

${\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)^{2}\phi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)-2T\left({\frac {a}{\sqrt {1+b^{2}}}},{\frac {1}{\sqrt {1+2b^{2}}}}\right)}$ 

${\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)^{2}\phi (x)\,dx={\frac {2b}{\sqrt {1+b^{2}}}}\phi ({\tfrac {a}{t}})\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}{\sqrt {1+2b^{2}}}}}\right)}$ ^{[nb 4]}

${\displaystyle \int _{-\infty }^{\infty }\Phi (bx)^{2}\phi (x)\,dx=\pi ^{-1}\arctan {\sqrt {1+2b^{2}}}}$ 

${\displaystyle \int _{-\infty }^{\infty }x\phi (x)\Phi (bx)\,dx=\int _{-\infty }^{\infty }x\phi (x)\Phi (bx)^{2}\,dx={\frac {b}{\sqrt {2\pi (1+b^{2})}}}}$ 

${\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)\phi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)}$ 

${\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)\phi (x)\,dx={\tfrac {b}{t}}\phi ({\tfrac {a}{t}}),\qquad t={\sqrt {1+b^{2}}}}$ 

${\displaystyle \int _{0}^{\infty }x\Phi (a+bx)\phi (x)\,dx={\tfrac {b}{t}}\phi ({\tfrac {a}{t}})\Phi (-{\tfrac {ab}{t}})+(2\pi )^{-1/2}\Phi (a),\qquad t={\sqrt {1+b^{2}}}}$ 

${\displaystyle \int _{-\infty }^{\infty }\ln(x^{2}){\tfrac {1}{\sigma }}\phi \left({\tfrac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036}$ 

References

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. {{cite book}}: Invalid |ref=harv (help)

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