List of integrals of Gaussian functions

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In the expressions in this article,

ϕ(x)=12πe−12x2

is the standard normal probability density function,

Φ(x)=∫−∞xϕ(t)dt=12(1+erf⁡(x2))

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

T(h,a)=ϕ(h)∫0aϕ(hx)1+x2dx

is Owen's T function.

Owen[1] has an extensive list of Gaussian-type integrals; only a subset is given below.

## Indefinite integrals

* ∫ϕ(x)dx=Φ(x)+C
* ∫xϕ(x)dx=−ϕ(x)+C
* ∫x2ϕ(x)dx=Φ(x)−xϕ(x)+C
* ∫x2k+1ϕ(x)dx=−ϕ(x)∑j=0k(2k)!!(2j)!!x2j+C[2]
* ∫x2k+2ϕ(x)dx=−ϕ(x)∑j=0k(2k+1)!!(2j+1)!!x2j+1+(2k+1)!!Φ(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.

* ∫ϕ(x)2dx=12πΦ(x2)+C
* ∫ϕ(x)ϕ(a+bx)dx=1tϕ(at)Φ(tx+at)+C,t=1+b2[3]
* ∫xϕ(a+bx)dx=−1b2(ϕ(a+bx)+aΦ(a+bx))+C
* ∫x2ϕ(a+bx)dx=1b3((a2+1)Φ(a+bx)+(a−bx)ϕ(a+bx))+C
* ∫ϕ(a+bx)ndx=1bn(2π)n−1Φ(n(a+bx))+C
* ∫Φ(a+bx)dx=1b((a+bx)Φ(a+bx)+ϕ(a+bx))+C
* ∫xΦ(a+bx)dx=12b2((b2x2−a2−1)Φ(a+bx)+(bx−a)ϕ(a+bx))+C
* ∫x2Φ(a+bx)dx=13b3((b3x3+a3+3a)Φ(a+bx)+(b2x2−abx+a2+2)ϕ(a+bx))+C
* ∫xnΦ(x)dx=1n+1((xn+1−nxn−1)Φ(x)+xnϕ(x)+n(n−1)∫xn−2Φ(x)dx)+C
* ∫xϕ(x)Φ(a+bx)dx=btϕ(at)Φ(xt+abt)−ϕ(x)Φ(a+bx)+C,t=1+b2
* ∫Φ(x)2dx=xΦ(x)2+2Φ(x)ϕ(x)−1πΦ(x2)+C
* ∫ecxϕ(bx)ndx=ec22nb2bn(2π)n−1Φ(b2xn−cbn)+C,b≠0,n>0

## Definite integrals

* ∫−∞∞x2ϕ(x)ndx=1n3(2π)n−1
* ∫−∞0ϕ(ax)Φ(bx)dx=12π|a|(π2−arctan⁡(b|a|))
* ∫0∞ϕ(ax)Φ(bx)dx=12π|a|(π2+arctan⁡(b|a|))
* ∫0∞xϕ(x)Φ(bx)dx=122π(1+b1+b2)
* ∫0∞x2ϕ(x)Φ(bx)dx=14+12π(b1+b2+arctan⁡(b))
* ∫−∞∞xϕ(x)2Φ(x)dx=14π3
* ∫0∞Φ(bx)2ϕ(x)dx=12π(arctan⁡(b)+arctan⁡1+2b2)
* ∫−∞∞Φ(a+bx)2ϕ(x)dx=Φ(a1+b2)−2T(a1+b2,11+2b2)
* ∫−∞∞xΦ(a+bx)2ϕ(x)dx=2b1+b2ϕ(at)Φ(a1+b21+2b2)[4]
* ∫−∞∞Φ(bx)2ϕ(x)dx=1πarctan⁡1+2b2
* ∫−∞∞xϕ(x)Φ(bx)dx=∫−∞∞xϕ(x)Φ(bx)2dx=b2π(1+b2)
* ∫−∞∞Φ(a+bx)ϕ(x)dx=Φ(a1+b2)
* ∫−∞∞xΦ(a+bx)ϕ(x)dx=btϕ(at),t=1+b2
* ∫0∞xΦ(a+bx)ϕ(x)dx=btϕ(at)Φ(−abt)+12πΦ(a),t=1+b2
* ∫−∞∞ln⁡(x2)1σϕ(xσ)dx=ln⁡(σ2)−γ−ln⁡2≈ln⁡(σ2)−1.27036

## References

1. ↑ Owen 1980.
2. ↑ (Patel Read) lists this integral above without the minus sign, which is an error. See calculation by WolframAlpha.
3. ↑ (Patel Read) report this integral with error, see WolframAlpha.
4. ↑ (Patel Read) report this integral incorrectly by omitting x from the integrand.

* 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.

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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