List of integrals of Gaussian functions

In these expressions \phi(x) = \tfrac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} is the standard normal probability density function, and \textstyle \Phi(x) = \int_{-\infty}^x \phi(t)dt = \frac12\big(1 %2B \operatorname{erf}\big(\frac{x}{\sqrt{2}}\big)\big) is the corresponding cumulative distribution function (where erf is the error function).

Indefinite integrals

\int \phi(x) \, dx = \Phi(x) %2B C
\int x \phi(x) \, dx = -\phi(x) %2B C
\int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) %2B C
\int x^{2k%2B1} \phi(x) \, dx = -\phi(x) \sum_{j=0}^k \frac{(2k)!!}{(2j)!!}x^{2j} %2B C [nb 1]
\int x^{2k%2B2} \phi(x) \, dx = -\phi(x)\sum_{j=0}^k\frac{(2k%2B1)!!}{(2j%2B1)!!}x^{2j%2B1} %2B (2k%2B1)!!\,\Phi(x) %2B 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)
\int \phi(x)^2 \, dx = \tfrac{1}{2\sqrt{\pi}} \Phi(x\sqrt{2}) %2B C
\int \phi(x)\phi(a %2B bx) \, dx = \tfrac{1}{t}\phi(a/t)\Phi(tx %2B ab/t) %2B C, \quad t = \sqrt{1%2Bb^2} [nb 2]
\int x\phi(a%2Bbx) \, dx = -\tfrac{1}{b^2}\phi(a%2Bbx) - \tfrac{a}{b^2}\Phi(a%2Bbx) %2B C
\int x^2\phi(a%2Bbx) \, dx = \tfrac{a^2%2B1}{b^3}\Phi(a%2Bbx) %2B \frac{a-bx}{b^3}\phi(a%2Bbx) %2B C
\int \phi(a%2Bbx)^n \, dx = \frac{(2\pi)^{-(n-1)/2}}{b\sqrt{n}} \Phi\big(\sqrt{n}(a%2Bbx)\big) %2B C
\int \Phi(a%2Bbx) \, dx = \tfrac{1}{b}(a%2Bbx)\Phi(a%2Bbx) %2B \tfrac{1}{b}\phi(a%2Bbx) %2B C
\int x\Phi(a%2Bbx) \, dx = \tfrac{1}{2b^2}\big((b^2x^2 - a^2 - 1)\Phi(a%2Bbx) %2B (bx-a)\phi(a%2Bbx)\big) %2B C
\int x^2\Phi(a%2Bbx) \, dx = \tfrac{1}{3b^3}\big((b^3x^3 %2B a^3 %2B 3a)\Phi(a%2Bbx) %2B (b^2x^2-abx%2Ba^2%2B2)\phi(a%2Bbx)\big) %2B C
\int x^n \Phi(x) \, dx = \tfrac{1}{n%2B1}\Big( (x^{n%2B1}-nx^{n-1})\Phi(x) %2B x^n\phi(x) %2B n(n-1)\int x^{n-2}\Phi(x)\,dx \Big) %2B C
\int x\phi(x)\Phi(a%2Bbx) \, dx = \tfrac{b}{t}\phi(a/t)\Phi(xt %2B ab/t) - \phi(x)\Phi(a%2Bbx) %2B C, \quad t = \sqrt{1%2Bb^2}
\int \Phi(x)^2 \, dx = x \Phi(x)^2 %2B 2\Phi(x)\phi(x) - \tfrac{1}{\sqrt{\pi}}\Phi(x\sqrt{2}) %2B C
\int e^{cx}\phi(bx)^n \, dx = \frac{1}{b\sqrt{n(2\pi)^{n-1}}}e^{c^2/(2nb^2)}\Phi(bx\sqrt{n} - \tfrac{c}{b\sqrt{n}}) %2B C, \quad b\ne0, n>0

Definite integrals

\begin{align} & \int_{-\infty}^\infty x^2\phi(x)^n \, \, dx = \Big(n^{3/2}(2\pi)^{(n-1)/2}\Big)^{-1} \\ & \int_{-\infty}^0 \phi(ax)\Phi(bx)dx = (2\pi a)^{-1}\arctan(a/b) \\ & \int_0^{\infty} \phi(ax)\Phi(bx) \, dx = (2\pi a)^{-1}\big(\tfrac{\pi}{2} - \arctan(b/a)\big) \\ & \int_0^\infty x\phi(x)\Phi(bx) \, dx = \frac{1}{2\sqrt{2\pi}} \bigg( 1 %2B \frac{b}{\sqrt{1%2Bb^2}} \bigg) \\ & \int_0^\infty x^2\phi(x)\Phi(bx) \, dx = \frac14 %2B \frac{1}{2\pi}\bigg( \frac{b}{1%2Bb^2} %2B \arctan b \bigg) \\ & \int x \phi(x)^2\Phi(x) \, dx = \frac{1}{4\pi\sqrt{3}} \\ & \int_0^\infty \Phi(bx)^2 \phi(x) \, dx = (2\pi)^{-1}\big( \arctan b %2B \arctan \sqrt{1%2B2b^2} \big) \\ & \int_{-\infty}^\infty \Phi(bx)^2 \phi(x) \, dx = \pi^{-1}\arctan \sqrt{1%2B2b^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%2Bb^2)}} \\ & \int_{-\infty}^\infty \Phi(a%2Bbx)\phi(x) \, dx = \Phi\big(a/\sqrt{1%2Bb^2}\big) \\ & \int_{-\infty}^\infty x\Phi(a%2Bbx)\phi(x) \, dx = (b/t)\phi(a/t), \quad t = \sqrt{1%2Bb^2} \\ & \int_0^\infty x\Phi(a%2Bbx)\phi(x) \, dx = (b/t)\phi(a/t)\Phi(-ab/t) %2B (2\pi)^{-1/2}\Phi(a), \quad t = \sqrt{1%2Bb^2} \\ & \int_{-\infty}^\infty \ln(x^2) \tfrac{1}{\sigma}\phi\big(\tfrac{x}{\sigma}\big) \, dx = \ln(\sigma^2) - \gamma - \ln 2 \approx \ln(\sigma^2) - 1.27036 \end{align}

References

  1. ^ Patel & Read (1996) list this integral without the minus sign, which is an error. See calculation by WolframAlpha
  2. ^ Patel & Read (1996) report this integral with error, see WolframAlpha