List of integrals of Gaussian functions

Short description: none

In the expressions in this article,

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

is the standard normal probability density function,

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

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

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

is Owen's T function.

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

Indefinite integrals

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

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.

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

Definite integrals

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

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.