List Of Integrals Of Gaussian Functions

In the expressions in this article,

:$\phi(x) = \frace^$

is the standard normal probability density function,

:$\Phi(x) = \int_^x \phi(t) \, dt = \frac\left(1 + \operatorname\left(\frac\right)\right)$

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

:$T(h,a) = \phi(h)\int_0^a \frac \, dx$

is Owen's T function.

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

Indefinite integrals

:$\int \phi(x) \, dx = \Phi(x) + C$
:$\int x \phi(x) \, dx = -\phi(x) + C$
:$\int x^2 \phi(x) \, dx = \Phi(x) - x\phi(x) + C$
:$\int x^ \phi(x) \, dx = -\phi(x) \sum_^k \fracx^ + C$ lists this integral above without the minus sign, which is an error. See calculation b
WolframAlpha
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:$\int x^ \phi(x) \, dx = -\phi(x)\sum_^k\fracx^ + (2k+1)!!\,\Phi(x) + C$

In these 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 .

: $\int \phi(x)^2 \, dx = \frac \Phi\left(x\sqrt\right) + C$
: $\int \phi(x)\phi(a + bx) \, dx = \frac\phi\left(\frac\right)\Phi\left(tx + \frac\right) + C, \qquad t = \sqrt$ report this integral with error, se
WolframAlpha
/ref>
: $\int x\phi(a+bx) \, dx = -\frac\left (\phi(a+bx) + a\Phi(a+bx)\right) + C$
: $\int x^2\phi(a+bx) \, dx = \frac \left ((a^2+1)\Phi(a+bx) + (a-bx)\phi(a+bx) \right ) + C$
: $\int \phi(a+bx)^n \, dx = \frac \Phi\left(\sqrt(a+bx)\right) + C$
: $\int \Phi(a+bx) \, dx = \frac \left ((a+bx)\Phi(a+bx) + \phi(a+bx)\right) + C$
: $\int x\Phi(a+bx) \, dx = \frac\left((b^2x^2 - a^2 - 1)\Phi(a+bx) + (bx-a)\phi(a+bx)\right) + C$
: $\int x^2\Phi(a+bx) \, dx = \frac\left((b^3x^3 + a^3 + 3a)\Phi(a+bx) + (b^2x^2-abx+a^2+2)\phi(a+bx)\right) + C$
: $\int x^n \Phi(x) \, dx = \frac\left( \left (x^-nx^ \right )\Phi(x) + x^n\phi(x) + n(n-1)\int x^\Phi(x)\,dx \right) + C$
: $\int x\phi(x)\Phi(a+bx) \, dx = \frac\phi\left(\frac\right)\Phi\left(xt + \frac\right) - \phi(x)\Phi(a+bx) + C, \qquad t = \sqrt$
: $\int \Phi(x)^2 \, dx = x \Phi(x)^2 + 2\Phi(x)\phi(x) - \frac\Phi\left(x\sqrt\right) + C$
: $\int e^\phi(bx)^n \, dx = \frac\Phi \left (\frac \right ) + C, \qquad b\ne 0, n>0$

Definite integrals

: $\int_^\infty x^2\phi(x)^n \, dx = \frac$
: $\int_^0 \phi(ax)\Phi(bx)dx = \frac\left(\frac-\arctan\left(\frac\right)\right)$
: $\int_0^ \phi(ax)\Phi(bx) \, dx = \frac\left(\frac + \arctan\left(\frac\right)\right)$
: $\int_0^\infty x\phi(x)\Phi(bx) \, dx = \frac \left( 1 + \frac \right)$
: $\int_0^\infty x^2\phi(x)\Phi(bx) \, dx = \frac + \frac \left(\frac + \arctan(b) \right)$
: $\int_^\infty x \phi(x)^2\Phi(x) \, dx = \frac$
: $\int_0^\infty \Phi(bx)^2 \phi(x) \, dx = \frac\left( \arctan(b) + \arctan \sqrt \right)$
: $\int_^\infty \Phi(a+bx)^2 \phi(x) \,dx = \Phi\left( \frac \right)-2T\left( \frac, \frac \right)$
: $\int_^ x \Phi(a+bx)^2 \phi(x) \,dx = \frac \phi\left(\frac\right) \Phi\left(\frac\right)$ report this integral incorrectly by omitting ''x'' from the integrand
: $\int_^\infty \Phi(bx)^2 \phi(x) \, dx = \frac\arctan \sqrt$
: $\int_^\infty x\phi(x)\Phi(bx) \, dx = \int_^\infty x\phi(x)\Phi(bx)^2 \, dx = \frac$
: $\int_^\infty \Phi(a+bx)\phi(x) \, dx = \Phi\left(\frac\right)$
: $\int_^\infty x\Phi(a+bx)\phi(x) \, dx = \frac\phi\left(\frac\right), \qquad t = \sqrt$
: $\int_0^\infty x\Phi(a+bx)\phi(x) \, dx =\frac\phi\left(\frac\right)\Phi\left(-\frac\right) + \frac\Phi(a), \qquad t = \sqrt$
: $\int_^\infty \ln(x^2) \frac\phi\left(\frac\right) \, dx = \ln(\sigma^2) - \gamma - \ln 2 \approx \ln(\sigma^2) - 1.27036$

References

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{{DEFAULTSORT:Integrals of Gaussian functions
Gaussian functions
Gaussian function