Some Handy Integrals - Colby College

1 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.

2 Even and odd Integrals . To determine if a function is even, check to see if f(x) = f(-x). For an odd function, f(x) = f(-x). some functions are neither odd nor even. For example, f(x) = x is odd, f(x) = x2 is even, and f(x) = x + x2 is neither odd nor even. The following multiplication rules hold: even*even = even odd*odd =even odd*even = odd Consider the integral of f(x) = e ax2, Figure 1a. The function is even so that - = 2 0. Next consider g(x) = x, which is odd, giving [g(x) f(x)] = x e ax2 as overall odd (Figure 1b). The integral is zero for the product function. x y 0 even Integrals add x y 0 odd Integrals cancel