In the expressions in this article,
![{\displaystyle \varphi (x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}x^{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3123d8dd4c3386afe9fac119fed2cfaf7ce9f336)
is the standard normal probability density function,
![{\displaystyle \Phi (x)=\int _{-\infty }^{x}\varphi (t)\,dt={\frac {1}{2}}\left(1+\operatorname {erf} \left({\frac {x}{\sqrt {2}}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b8b0cc5439fca2b5d43257431fb5c252090d653)
is the corresponding cumulative distribution function (where erf is the error function), and
![{\displaystyle T(h,a)=\varphi (h)\int _{0}^{a}{\frac {\varphi (hx)}{1+x^{2}}}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bdb303a2c8dd6e9af6810406c33c0076988cd47)
is Owen's T function.
Owen[1] has an extensive list of Gaussian-type integrals; only a subset is given below.
Indefinite integrals
![{\displaystyle \int \varphi (x)\,dx=\Phi (x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/00e9ad95ceeb3812010d01ec73811d0c4c623fc8)
![{\displaystyle \int x\varphi (x)\,dx=-\varphi (x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9b420bb477db54fa10a0c7accfb2c42a1240c55)
![{\displaystyle \int x^{2}\varphi (x)\,dx=\Phi (x)-x\varphi (x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1d075a5ba9492e9e3fd826c7de650ac85bd90a)
[2] ![{\displaystyle \int x^{2k+2}\varphi (x)\,dx=-\varphi (x)\sum _{j=0}^{k}{\frac {(2k+1)!!}{(2j+1)!!}}x^{2j+1}+(2k+1)!!\,\Phi (x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/517373392fa55a638bd5d74b232397c772bf90d4)
In the previous two 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 \varphi (x)^{2}\,dx={\frac {1}{2{\sqrt {\pi }}}}\Phi \left(x{\sqrt {2}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59056de925da7d04548f907b5b8d39196b7e35ab)
[3] ![{\displaystyle \int x\varphi (a+bx)\,dx=-{\frac {1}{b^{2}}}\left(\varphi (a+bx)+a\Phi (a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e5e6a0122d9dc4538a699b7487828bcd990ab0)
![{\displaystyle \int x^{2}\varphi (a+bx)\,dx={\frac {1}{b^{3}}}\left((a^{2}+1)\Phi (a+bx)+(a-bx)\varphi (a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7acfae876d626af29ac08e39512e408e22489a7)
![{\displaystyle \int \varphi (a+bx)^{n}\,dx={\frac {1}{b{\sqrt {n(2\pi )^{n-1}}}}}\Phi \left({\sqrt {n}}(a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f974b00214a1c1ce365fd8c7bb08cc0bbe7aac3)
![{\displaystyle \int \Phi (a+bx)\,dx={\frac {1}{b}}\left((a+bx)\Phi (a+bx)+\varphi (a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79dd1e621333623a19f86b12d4a37153d0b126ee)
![{\displaystyle \int x\Phi (a+bx)\,dx={\frac {1}{2b^{2}}}\left((b^{2}x^{2}-a^{2}-1)\Phi (a+bx)+(bx-a)\varphi (a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/074542c127feb9d7b12ec7cb67f9ef6d48af5520)
![{\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)\varphi (a+bx)\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d677de575e2fbb23d843bcfb11abe7027f9a9f)
![{\displaystyle \int x^{n}\Phi (x)\,dx={\frac {1}{n+1}}\left(\left(x^{n+1}-nx^{n-1}\right)\Phi (x)+x^{n}\varphi (x)+n(n-1)\int x^{n-2}\Phi (x)\,dx\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d66695d20ec60d3bb6fd2a4f49f78052e98462e)
![{\displaystyle \int x\varphi (x)\Phi (a+bx)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right)\Phi \left(xt+{\frac {ab}{t}}\right)-\varphi (x)\Phi (a+bx)+C,\qquad t={\sqrt {1+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/259307199e7d339cb38124f35d495745dbb35b77)
![{\displaystyle \int \Phi (x)^{2}\,dx=x\Phi (x)^{2}+2\Phi (x)\varphi (x)-{\frac {1}{\sqrt {\pi }}}\Phi \left(x{\sqrt {2}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/507804fce0cdb7a37dc01cec734f1b86201da443)
![{\displaystyle \int e^{cx}\varphi (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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5135ddead869c04885393acce177267cc06d11d9)
Definite integrals
![{\displaystyle \int _{-\infty }^{\infty }x^{2}\varphi (x)^{n}\,dx={\frac {1}{\sqrt {n^{3}(2\pi )^{n-1}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93dbf9121f7d030e8a3bb13b156a799590ef0001)
![{\displaystyle \int _{-\infty }^{\infty }\varphi (x)\varphi (a+bx)\,dx={\frac {1}{\sqrt {1+b^{2}}}}\varphi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4ef5bfff4592b33c721b96300540d3ef47f41d)
![{\displaystyle \int _{-\infty }^{0}\varphi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}-\arctan \left({\frac {b}{|a|}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd162aef26bd058b784cacb642f547cdb1c7132e)
![{\displaystyle \int _{0}^{\infty }\varphi (ax)\Phi (bx)\,dx={\frac {1}{2\pi |a|}}\left({\frac {\pi }{2}}+\arctan \left({\frac {b}{|a|}}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bffc7ad2e21924d78791a44415b8cd540072105d)
![{\displaystyle \int _{0}^{\infty }x\varphi (x)\Phi (bx)\,dx={\frac {1}{2{\sqrt {2\pi }}}}\left(1+{\frac {b}{\sqrt {1+b^{2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a84c69819ca7c90f999bba8ec96874f0e3b01590)
![{\displaystyle \int _{0}^{\infty }x^{2}\varphi (x)\Phi (bx)\,dx={\frac {1}{4}}+{\frac {1}{2\pi }}\left({\frac {b}{1+b^{2}}}+\arctan(b)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e028bf541e019fa1c59cb33b630eaee58de304e)
![{\displaystyle \int _{-\infty }^{\infty }x\varphi (x)^{2}\Phi (x)\,dx={\frac {1}{4\pi {\sqrt {3}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6eb81a0d173b1fab00f1e9f02b6628b2512bce8)
![{\displaystyle \int _{0}^{\infty }\Phi (bx)^{2}\varphi (x)\,dx={\frac {1}{2\pi }}\left(\arctan(b)+\arctan {\sqrt {1+2b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/203f6dd5132646cb39c245a66ed02e9ae47ee742)
![{\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)^{2}\varphi (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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c554557e366da513e0b0c35b597051011db322e4)
[4] ![{\displaystyle \int _{-\infty }^{\infty }\Phi (bx)^{2}\varphi (x)\,dx={\frac {1}{\pi }}\arctan {\sqrt {1+2b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7a833662a5fec995e7a201e497d2f47395999f)
![{\displaystyle \int _{-\infty }^{\infty }x\varphi (x)\Phi (bx)\,dx=\int _{-\infty }^{\infty }x\varphi (x)\Phi (bx)^{2}\,dx={\frac {b}{\sqrt {2\pi (1+b^{2})}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/510080f50eeaae737a5873ed1d3e9a86acbf96b6)
![{\displaystyle \int _{-\infty }^{\infty }\Phi (a+bx)\varphi (x)\,dx=\Phi \left({\frac {a}{\sqrt {1+b^{2}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eca87d657140563119754cbe6effecb06071b3c)
![{\displaystyle \int _{-\infty }^{\infty }x\Phi (a+bx)\varphi (x)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right),\qquad t={\sqrt {1+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e85700affa317250e9640ea48ebd36aed27cf37)
![{\displaystyle \int _{0}^{\infty }x\Phi (a+bx)\varphi (x)\,dx={\frac {b}{t}}\varphi \left({\frac {a}{t}}\right)\Phi \left(-{\frac {ab}{t}}\right)+{\frac {1}{\sqrt {2\pi }}}\Phi (a),\qquad t={\sqrt {1+b^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df2815466d2a29b4da23f104f48425153ad54c00)
![{\displaystyle \int _{-\infty }^{\infty }\ln(x^{2}){\frac {1}{\sigma }}\varphi \left({\frac {x}{\sigma }}\right)\,dx=\ln(\sigma ^{2})-\gamma -\ln 2\approx \ln(\sigma ^{2})-1.27036}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8895aa56088f04665329e00ec071a5af3c4630)
References
- Owen, D. (1980). "A table of normal integrals". Communications in Statistics: Simulation and Computation. B9 (4): 389–419. doi:10.1080/03610918008812164.
- Patel, Jagdish K.; Read, Campbell B. (1996). Handbook of the normal distribution (2nd ed.). CRC Press. ISBN 0-8247-9342-0.