List of integrals of Gaussian functions

Short description: none

In the expressions in this article,

[math]\displaystyle{ \phi(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2} x^2} }[/math]

is the standard normal probability density function,

[math]\displaystyle{ \Phi(x) = \int_{-\infty}^x \phi(t) \, dt = \frac{1}{2}\left(1 + \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)\right) }[/math]

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

[math]\displaystyle{ T(h,a) = \phi(h)\int_0^a \frac{\phi(hx)}{1+x^2} \, dx }[/math]

is Owen's T function.

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

Indefinite integrals

• [math]\displaystyle{ \int \phi(x) \, dx = \Phi(x) + C }[/math]
• [math]\displaystyle{ \int x \phi(x) \, dx = -\phi(x) + C }[/math]
• [math]\displaystyle{ \int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) + C }[/math]
• [math]\displaystyle{ \int x^{2k+1} \phi(x) \, dx = -\phi(x) \sum_{j=0}^k \frac{(2k)!!}{(2j)!!}x^{2j} + C }[/math]
• [math]\displaystyle{ \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 }[/math]

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.

• [math]\displaystyle{ \int \phi(x)^2 \, dx = \frac{1}{2\sqrt{\pi}} \Phi\left(x\sqrt{2}\right) + C }[/math]
• [math]\displaystyle{ \int \phi(x)\phi(a + bx) \, dx = \frac{1}{t}\phi\left(\frac{a}{t}\right)\Phi\left(tx + \frac{a}{t}\right) + C, \qquad t = \sqrt{1+b^2} }[/math]
• [math]\displaystyle{ \int x\phi(a+bx) \, dx = -\frac{1}{b^2}\left (\phi(a+bx) + a\Phi(a+bx)\right) + C }[/math]
• [math]\displaystyle{ \int x^2\phi(a+bx) \, dx = \frac{1}{b^3} \left ((a^2+1)\Phi(a+bx) + (a-bx)\phi(a+bx) \right ) + C }[/math]
• [math]\displaystyle{ \int \phi(a+bx)^n \, dx = \frac{1}{b\sqrt{n(2\pi)^{n-1}}} \Phi\left(\sqrt{n}(a+bx)\right) + C }[/math]
• [math]\displaystyle{ \int \Phi(a+bx) \, dx = \frac{1}{b} \left ((a+bx)\Phi(a+bx) + \phi(a+bx)\right) + C }[/math]
• [math]\displaystyle{ \int x\Phi(a+bx) \, dx = \frac{1}{2b^2}\left((b^2x^2 - a^2 - 1)\Phi(a+bx) + (bx-a)\phi(a+bx)\right) + C }[/math]
• [math]\displaystyle{ \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)\phi(a+bx)\right) + C }[/math]
• [math]\displaystyle{ \int x^n \Phi(x) \, dx = \frac{1}{n+1}\left( \left (x^{n+1}-nx^{n-1} \right )\Phi(x) + x^n\phi(x) + n(n-1)\int x^{n-2}\Phi(x)\,dx \right) + C }[/math]
• [math]\displaystyle{ \int x\phi(x)\Phi(a+bx) \, dx = \frac{b}{t}\phi\left(\frac{a}{t}\right)\Phi\left(xt + \frac{ab}{t}\right) - \phi(x)\Phi(a+bx) + C, \qquad t = \sqrt{1+b^2} }[/math]
• [math]\displaystyle{ \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 }[/math]
• [math]\displaystyle{ \int e^{cx}\phi(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\gt 0 }[/math]

Definite integrals

• [math]\displaystyle{ \int_{-\infty}^\infty x^2\phi(x)^n \, dx = \frac{1}{\sqrt{n^3(2\pi)^{n-1}}} }[/math]
• [math]\displaystyle{ \int_{-\infty}^0 \phi(ax)\Phi(bx) \, dx = \frac{1}{2\pi |a|}\left(\frac{\pi}{2}-\arctan\left(\frac{b}{|a|}\right)\right) }[/math]
• [math]\displaystyle{ \int_0^{\infty} \phi(ax)\Phi(bx) \, dx = \frac{1}{2\pi |a|}\left(\frac{\pi}{2} + \arctan\left(\frac{b}{|a|}\right)\right) }[/math]
• [math]\displaystyle{ \int_0^\infty x\phi(x)\Phi(bx) \, dx = \frac{1}{2\sqrt{2\pi}} \left( 1 + \frac{b}{\sqrt{1+b^2}} \right) }[/math]
• [math]\displaystyle{ \int_0^\infty x^2\phi(x)\Phi(bx) \, dx = \frac{1}{4} + \frac{1}{2\pi} \left(\frac{b}{1+b^2} + \arctan(b) \right) }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty x \phi(x)^2\Phi(x) \, dx = \frac{1}{4\pi\sqrt{3}} }[/math]
• [math]\displaystyle{ \int_0^\infty \Phi(bx)^2 \phi(x) \, dx = \frac{1}{2\pi}\left( \arctan(b) + \arctan \sqrt{1+2b^2} \right) }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty \Phi(a+bx)^2 \phi(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) }[/math]
• [math]\displaystyle{ \int_{-\infty}^{\infty} 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) }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty \Phi(bx)^2 \phi(x) \, dx = \frac{1}{\pi}\arctan \sqrt{1+2b^2} }[/math]
• [math]\displaystyle{ \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+b^2)}} }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty \Phi(a+bx)\phi(x) \, dx = \Phi\left(\frac{a}{\sqrt{1+b^2}}\right) }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty x\Phi(a+bx)\phi(x) \, dx = \frac{b}{t}\phi\left(\frac{a}{t}\right), \qquad t = \sqrt{1+b^2} }[/math]
• [math]\displaystyle{ \int_0^\infty x\Phi(a+bx)\phi(x) \, dx =\frac{b}{t}\phi\left(\frac{a}{t}\right)\Phi\left(-\frac{ab}{t}\right) + \frac{1}{\sqrt{2\pi}}\Phi(a), \qquad t = \sqrt{1+b^2} }[/math]
• [math]\displaystyle{ \int_{-\infty}^\infty \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 }[/math]

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.