List of integrals of Gaussian functions

Template:SHORTDESC: In the expressions in this article,

$$\phi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}$$

is the standard normal probability density function,

$$\Phi (x)=\underset{-\mathrm{\infty }}{\overset{x}{\int }}\phi (t)dt=\frac{1}{2}[1+\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)]$$

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

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

is Owen's T function.

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

Indefinite integrals

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

In 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.

• $\int \phi (x{)}^{2}dx=\frac{1}{2\sqrt{\pi }}\Phi \left(x\sqrt{2}\right)+C$
• $\int \phi (x)\phi (a+bx)dx=\frac{1}{t}\phi \left(\frac{a}{t}\right)\Phi (tx+\frac{ab}{t})+C,t=\sqrt{1+b^2}$
• $\int x\phi (a+bx)dx=-\frac{1}{b^2}[\phi (a+bx)+a\Phi (a+bx)]+C$
• $\int x^2\phi (a+bx)dx=\frac{1}{b^3}[(a^2+1)\Phi (a+bx)+(a-bx)\phi (a+bx)]+C$
• $\int \phi (a+bx{)}^{n}dx=\frac{1}{b\sqrt{n{\left(2\pi \right)}^{n-1}}}\Phi (\sqrt{n}(a+bx))+C$
• $\int \Phi (a+bx)dx=\frac{1}{b}[(a+bx)\Phi (a+bx)+\phi (a+bx)]+C$
• $\int x\Phi (a+bx)dx=\frac{1}{2b^2}[(b^2{x}^2-a^2-1)\Phi (a+bx)+(bx-a)\phi (a+bx)]+C$
• $\int x^2\Phi (a+bx)dx=\frac{1}{3b^3}[(b^3{x}^3+a^3+3a)\Phi (a+bx)+(b^2{x}^2-abx+a^2+2)\phi (a+bx)]+C$
• $\int x^n\Phi (x)dx=\frac{1}{n+1}[(x^{n+1}-n{x}^{n-1})\Phi (x)+x^n\phi (x)+n(n-1)\int x^{n-2}\Phi (x)dx]+C$
• $\int x\phi (x)\Phi (a+bx)dx=\frac{b}{t}\phi \left(\frac{a}{t}\right)\Phi (xt+\frac{ab}{t})-\phi (x)\Phi (a+bx)+C,t=\sqrt{1+b^2}$
• $\int \Phi (x{)}^{2}dx=x\Phi (x{)}^2+2\Phi (x)\phi (x)-\frac{1}{\sqrt{\pi }}\Phi \left(x\sqrt{2}\right)+C$
• $\int e^{cx}\phi (bx{)}^{n}dx=\frac{e^{\frac{c^2}{2n{b}^2}}}{b\sqrt{n{\left(2\pi \right)}^{n-1}}}\Phi \left(\frac{b^{2}xn-c}{b\sqrt{n}}\right)+C,b\ne 0,n>0$

Definite integrals

• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^2\phi (x{)}^{n}dx=\frac{1}{\sqrt{n^3{\left(2\pi \right)}^{n-1}}}$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\phi (x)\phi (a+bx)dx=\frac{1}{\sqrt{1+b^2}}\phi \left(\frac{a}{\sqrt{1+b^2}}\right)$
• $\underset{-\mathrm{\infty }}{\overset{0}{\int }}\phi (ax)\Phi (bx)dx=\frac{1}{2\pi |a|}(\frac{\pi }{2}-\arctan \left(\frac{b}{|a|}\right))$
• $\underset{0}{\overset{\mathrm{\infty }}{\int }}\phi (ax)\Phi (bx)dx=\frac{1}{2\pi |a|}(\frac{\pi }{2}+\arctan \left(\frac{b}{|a|}\right))$
• $\underset{0}{\overset{\mathrm{\infty }}{\int }}x\phi (x)\Phi (bx)dx=\frac{1}{2\sqrt{2\pi }}(1+\frac{b}{\sqrt{1+b^2}})$
• $\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^2\phi (x)\Phi (bx)dx=\frac{1}{4}+\frac{1}{2\pi }(\frac{b}{1+b^2}+\arctan (b))$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x\phi (x{)}^2\Phi (x)dx=\frac{1}{4\pi \sqrt{3}}$
• $\underset{0}{\overset{\mathrm{\infty }}{\int }}\Phi (bx{)}^2\phi (x)dx=\frac{1}{2\pi }(\arctan (b)+\arctan \sqrt{1+2b^2})$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\Phi (a+bx{)}^2\phi (x)dx=\Phi \left(\frac{a}{\sqrt{1+b^2}}\right)-2T(\frac{a}{\sqrt{1+b^2}},\frac{1}{\sqrt{1+2b^2}})$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x\Phi (a+bx{)}^2\phi (x)dx=\frac{2b}{\sqrt{1+b^2}}\phi \left(\frac{a}{t}\right)\Phi \left(\frac{a}{\sqrt{1+b^2}\sqrt{1+2b^2}}\right)$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\Phi (bx{)}^2\phi (x)dx=\frac{1}{\pi }\arctan \sqrt{1+2b^2}$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x\phi (x)\Phi (bx)dx=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x\phi (x)\Phi (bx{)}^{2}dx=\frac{b}{\sqrt{2\pi (1+b^2)}}$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\Phi (a+bx)\phi (x)dx=\Phi \left(\frac{a}{\sqrt{1+b^2}}\right)$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}x\Phi (a+bx)\phi (x)dx=\frac{b}{\sqrt{1+b^2}}\phi \left(\frac{a}{\sqrt{1+b^2}}\right),$
• $\underset{0}{\overset{\mathrm{\infty }}{\int }}x\Phi (a+bx)\phi (x)dx=\frac{b}{t}\phi \left(\frac{a}{t}\right)\Phi (-\frac{ab}{t})+\frac{1}{\sqrt{2\pi }}\Phi (a),t=\sqrt{1+b^2}$
• $\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\ln (x^2)\frac{1}{\sigma }\phi \left(\frac{x}{\sigma }\right)dx=\ln ({\sigma }^2)-\gamma -\ln 2\approx \ln ({\sigma }^2)-1.27036$