In these expressions,
\( {\displaystyle \phi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}} \)
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
\( T(h,a)=\phi (h)\int _{0}^{a}{\frac {\phi (hx)}{1+x^{2}}}\,dx \)
is Owen's T function.
Owen[nb 1] has an extensive list of Gaussian-type integrals; only a subset is given below.
Indefinite integrals
\( \int \phi (x)\,dx=\Phi (x)+C \)
\( \int x\phi (x)\,dx=-\phi (x)+C \)
\( \int x^{2}\phi (x)\,dx=\Phi (x)-x\phi (x)+C \)
\( \int x^{{2k+1}}\phi (x)\,dx=-\phi (x)\sum _{{j=0}}^{k}{\frac {(2k)!!}{(2j)!!}}x^{{2j}}+C[nb 2] \)
\( \int x^{{2k+2}}\phi (x)\,dx=-\phi (x)\sum _{{j=0}}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{{2j+1}}+(2k+1)!!\,\Phi (x)+C \)
In these integrals, n!! is the double factorial: for even n 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.
\( {\displaystyle \int \phi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C} \)
\( {\displaystyle \int \phi (x)\phi (a+bx)\,dx={\frac {1}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(tx+{\frac {ab}{t}}\right)+C,\qquad t={\sqrt {1+b^{2}}}} \) [nb 3]
\( {\displaystyle \int x\phi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\phi (a+bx)+a\Phi (a+bx)\right)+C} \)
\( {\displaystyle \int x^{2}\phi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\phi (a+bx)\right)+C} \)
\( \int \phi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{{n-1}}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C \)
\( {\displaystyle \int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\phi (a+bx)\right)+C} \)
\( {\displaystyle \int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\phi (a+bx)\right)+C} \)
\( {\displaystyle \int x^{2}\Phi (a+bx)\,dx={\frac {1}{3b^{3}}}\left((b^{3}x^{3}+a^{3}+3a)\Phi (a+bx)+(b^{2}x^{2}-abx+a^{2}+2)\phi (a+bx)\right)+C} \)
∫\( \int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{{n+1}}-nx^{{n-1}}\right)\Phi (x)+x^{n}\phi (x)+n(n-1)\int x^{{n-2}}\Phi (x)\,dx\right)+C \)
\( {\displaystyle \int x\phi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\phi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}} \)
\( {\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\phi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C} \)
\( \int e^{{cx}}\phi (bx)^{n}\,dx={\frac {e^{{{\frac {c^{2}}{2nb^{2}}}}}}{b{\sqrt {n(2\pi )^{{n-1}}}}}}\Phi \left({\frac {b^{2}xn-c}{b{\sqrt {n}}}}\right)+C,\qquad b\neq 0,n>0 \)
Definite integrals
∫\( \int _{{-\infty }}^{\infty }x^{2}\phi (x)^{n}\,dx={\frac {1}{{\sqrt {n^{3}(2\pi )^{{n-1}}}}}} \)
\( {\displaystyle \int _{-\infty }^{0}\phi (ax)\Phi (bx)dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}-\arctan \left({\frac {b}{|a|}}\right)\right)} \)
\( {\displaystyle \int _{0}^{\infty }\phi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}+\arctan \left({\frac {b}{|a|}}\right)\right)} \)
\( \int _{0}^{\infty }x\phi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{{\sqrt {1+b^{2}}}}}\right) \)
\( \int _{0}^{\infty }x^{2}\phi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right) \)
\( {\displaystyle \int _{0}^{\infty }x\phi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}} \)
\( {\displaystyle \int _{0}^{\infty }\Phi (bx)^{2}\phi (x)\,dx={\frac {1}{2\pi }}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)} \)
\( \int _{{-\infty }}^{\infty }\Phi (a+bx)^{2}\phi (x)\,dx=\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}}}\right)-2T\left({\frac {a}{{\sqrt {1+b^{2}}}}},{\frac {1}{{\sqrt {1+2b^{2}}}}}\right) \)
\( {\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)^{2}\phi (x)\,dx={\frac {2b}{\sqrt {1+b^{2}}}}\phi \left({\frac {a}{t}}\right)\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}{\sqrt {1+2b^{2}}}}}\right)} \) [nb 4]
\( {\displaystyle \int _{-\infty }^{\infty }\Phi (bx)^{2}\phi (x)\,dx={\frac {1}{\pi }}\arctan {\sqrt {1+2b^{2}}}} \)
\( \int _{{-\infty }}^{\infty }x\phi (x)\Phi (bx)\,dx=\int _{{-\infty }}^{\infty }x\phi (x)\Phi (bx)^{2}\,dx={\frac {b}{{\sqrt {2\pi (1+b^{2})}}}} \)
\( \int _{{-\infty }}^{\infty }\Phi (a+bx)\phi (x)\,dx=\Phi \left({\frac {a}{{\sqrt {1+b^{2}}}}}\right) \)
\( {\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)\phi (x)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right),\qquad t={\sqrt {1+b^{2}}}} \)
\( {\displaystyle \int _{0}^{\infty }x\Phi (a+bx)\phi (x)\,dx={\frac {b}{t}}\phi \left({\frac {a}{t}}\right)\Phi \left(-{\frac {ab}{t}}\right)+{\frac {1}{\sqrt {2\pi }}}\Phi (a),\qquad t={\sqrt {1+b^{2}}}} \)
\( {\displaystyle \int _{-\infty }^{\infty }\ln(x^{2}){\frac {1}{\sigma }}\phi \left({\frac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036} \)
References
Owen (1980)
Patel & Read (1996) lists this integral above without the minus sign, which is an error. See calculation by WolframAlpha
Patel & Read (1996) report this integral with error, see WolframAlpha
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: 389–419.
vte
Lists of integrals
Rational functions Irrational functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Inverse hyperbolic functions Exponential functions Logarithmic functions Gaussian functions Definite integrals
Undergraduate Texts in Mathematics
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