Some Handy Integrals

Some Handy Integrals Gaussian Functions 1 π½ 0 e–ax2 dx = 2 a 1 π½ 0 x2 e–ax2 dx = 4a a 3 π½ 0 x4 e–ax2 dx = 8a2 a 1·3·5···(2n–1) π½ 0 x2n e–ax2 dx = a 2n+1an  

1 dx = 2a

0 x e–ax

2

1 2a2 1 0 x5 e–ax2 dx = a3 n! 1 0 x2n+1 e–ax2 dx = 2 an+1

0 x3 e–ax

2

dx =

Exponential Functions n!   0 xn e–ax dx = n+1 a 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. even

y odd

y

x

0 x

0

integrals cancel

integrals add 2

2

(a). f(x) = e–ax

(b). [g(x) f(x)] = x e–ax

even

odd*even

Figure 1. 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–ax , Figure 1a. The function is even so that - = 20 . Next 2 consider g(x) = x, which is odd, giving [g(x) f(x)] = x e–ax as overall odd (Figure 1b). The integral is zero for the product function. 2

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