List of integrals of Gaussian functions

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,

$$\Phi(x) = \int_{-\infty}^x \varphi(t) \, dt = \frac{1}{2}\left(1 + \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\right)$$

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

$$ T(h,a) = \varphi(h)\int_0^a \frac{\varphi(hx)}{1+x^2} \, dx$$

is Owen's T function.

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

Indefinite integrals

 * $$\int \varphi(x) \, dx = \Phi(x) + C$$
 * $$\int x \varphi(x) \, dx = -\varphi(x) + C$$
 * $$\int x^2 \varphi(x) \, dx = \Phi(x) - x\varphi(x) + C$$
 * $$\int x^{2k+1} \varphi(x) \, dx = -\varphi(x) \sum_{j=0}^k \frac{(2k)!!}{(2j)!!}x^{2j} + C$$
 * $$\int x^{2k+2} \varphi(x) \, dx = -\varphi(x)\sum_{j=0}^k\frac{(2k+1)!!}{(2j+1)!!}x^{2j+1} + (2k+1)!!\,\Phi(x) + C$$

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 $0!! = (−1)!! = 1$.


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

Definite integrals

 * $$ \int_{-\infty}^\infty x^2\varphi(x)^n \, dx = \frac{1}{\sqrt{n^3(2\pi)^{n-1}}} $$
 * $$ \int_{-\infty}^\infty \varphi(x)\varphi(a+bx) \, dx = \frac{1}{\sqrt{1+b^2}}\varphi\left(\frac{a}{\sqrt{1+b^2}}\right) $$
 * $$\int_{-\infty}^0 \varphi(ax)\Phi(bx) \, dx = \frac{1}{2\pi |a|}\left(\frac{\pi}{2}-\arctan\left(\frac{b}{|a|}\right)\right) $$
 * $$\int_0^{\infty} \varphi(ax)\Phi(bx) \, dx = \frac{1}{2\pi |a|}\left(\frac{\pi}{2} + \arctan\left(\frac{b}{|a|}\right)\right) $$
 * $$ \int_0^\infty x\varphi(x)\Phi(bx) \, dx = \frac{1}{2\sqrt{2\pi}} \left( 1 + \frac{b}{\sqrt{1+b^2}} \right) $$
 * $$ \int_0^\infty x^2\varphi(x)\Phi(bx) \, dx = \frac{1}{4} + \frac{1}{2\pi} \left(\frac{b}{1+b^2} + \arctan(b) \right) $$
 * $$ \int_{-\infty}^\infty x \varphi(x)^2\Phi(x) \, dx = \frac{1}{4\pi\sqrt{3}} $$
 * $$ \int_0^\infty \Phi(bx)^2 \varphi(x) \, dx = \frac{1}{2\pi}\left( \arctan(b) + \arctan \sqrt{1+2b^2} \right) $$
 * $$ \int_{-\infty}^\infty \Phi(a+bx)^2 \varphi(x) \,dx = \Phi\left( \frac{a}{\sqrt{1+b^2}} \right)-2T\left( \frac{a}{\sqrt{1+b^2}}, \frac{1}{\sqrt{1+2b^2}} \right) $$
 * $$ \int_{-\infty}^{\infty} x \Phi(a+bx)^2 \varphi(x) \,dx = \frac{2b}{\sqrt{1+b^2}} \varphi\left(\frac{a}{t}\right) \Phi\left(\frac{a}{\sqrt{1+b^2}\sqrt{1+2b^2}}\right)$$
 * $$ \int_{-\infty}^\infty \Phi(bx)^2 \varphi(x) \, dx = \frac{1}{\pi}\arctan \sqrt{1+2b^2} $$
 * $$ \int_{-\infty}^\infty x\varphi(x)\Phi(bx) \, dx = \int_{-\infty}^\infty x\varphi(x)\Phi(bx)^2 \, dx = \frac{b}{\sqrt{2\pi(1+b^2)}} $$
 * $$ \int_{-\infty}^\infty \Phi(a+bx)\varphi(x) \, dx = \Phi\left(\frac{a}{\sqrt{1+b^2}}\right) $$
 * $$ \int_{-\infty}^\infty x\Phi(a+bx)\varphi(x) \, dx = \frac{b}{t}\varphi\left(\frac{a}{t}\right), \qquad t = \sqrt{1+b^2} $$
 * $$ \int_0^\infty x\Phi(a+bx)\varphi(x) \, dx =\frac{b}{t}\varphi\left(\frac{a}{t}\right)\Phi\left(-\frac{ab}{t}\right) + \frac{1}{\sqrt{2\pi}}\Phi(a), \qquad t = \sqrt{1+b^2} $$
 * $$ \int_{-\infty}^\infty \ln(x^2) \frac{1}{\sigma}\varphi\left(\frac{x}{\sigma}\right) \, dx = \ln(\sigma^2) - \gamma - \ln 2 \approx \ln(\sigma^2) - 1.27036 $$