Some Handy Integrals - Colby College

Colby College some Handy Integrals Gaussian Functions 0 e ax2 dx = 12 a 0 x e ax2 dx = 12a 0 x2 e ax2 dx = 14a a 0 x3 e ax2 dx = 12a2 0 x4 e ax2 dx = 38a2 a 0 x5 e ax2 dx = 1a3 0 x2n e ax2 dx = 1 3 5 (2n 1)2n+1an a 0 x2n+1 e ax2 dx = n!2 1an+1 Exponential Functions 0 xn e ax dx = n!an+1 Integrals from - to : Even and Odd Functions The integral of any even function taken between the limits - to is twice the integral from 0 to . The integral of any odd function between - and is equal to zero, see Figure 1. (a). f(x) = e ax2 (b). [g(x) f(x)] = x e ax2 even odd*even Figure 1.

Colby College Some Handy Integrals Gaussian Functions 0 2e–ax 2dx = 1 2 π a ½ 0 x e–ax dx = 1 2a 0 x 2 e–ax2 dx = 1 4a π a ½ 0 x 3 e–ax2 dx = 1 2a2 0 x 4 e–ax 2 dx = 3

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