Colby College Some Handy Integrals Gaussian Functions f dx f x e dx a f dx a f dx - PDF document

Colby College Some Handy Integrals Gaussian Functions   0 e – a x 2 dx = 1 2       π a ½   0 x e – a x 2 dx = 1 2a   0 x 2 e – a x 2 dx = 1 4a       π a ½   0 x 3 e – a x 2 dx = 1 2a 2   0 x 4 e – a x 2 dx = 3 8a 2       π a ½   0 x 5 e – a x 2 dx = 1 a 3   0 x 2n e – a x 2 dx = 1·3·5··· ( 2n – 1 ) 2 n+1 a n       π a ½   0 x 2n+1 e – a x 2 dx = n! 2       1 a n+1 Exponential Functions  0  x n e – ax d x = n! a n+1 I ntegrals from -  to  : Even and Odd Functions The i ntegra l of any even function take n 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 – a x 2 (b) . [ g (x) f (x) ] = x e – a x 2 even odd * even Figure 1. Even and odd integrals. T o determine if a function is even, check to see if f(x) = f( - x). For an odd function, f(x) = – f( - x). Some funct ions are neither odd nor even. For example, f(x) = x is odd, f(x) = x 2 is even, and f(x) = x + x 2 is neither odd nor even. T he following multipli cation rules hold: even*even = even odd*odd =e ven odd*even = odd C onsider the integral of f (x) = e – a x 2 , 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 – a x 2 a s overall odd (Figure 1 b ). The integral is zero for the product function. x y 0 even integrals add x y 0 odd integrals cancel