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+abt)+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
* ∫−∞∞φ(x)φ(a+bx)dx=11+b2φ(a1+b2)
* ∫−∞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=b1+b2φ(a1+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 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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