`int_(-a)^(a)f(x)dx= 2int_(0)^(a)f(x)dx,` if f is an even function
0, if f is an odd function.

Explanation

L.H.S. = `int_(-a)^(a) f(x) * dx `
` = int_(-a)^(0) f(x) * dx + int_(0)^(a) f(x) * dx `
` [:' - a lt 0 lt a] = I + int_(0)^(a) f (x) * dx ` ....(1)
Now, ` I = int _(-a)^(0) f (x) * dx `
Let ` x = - t" " :. dx = - dt `
When x = - a, t = a , when x = 0 , t = 0 ,
` I = int_(a)^(0)f(-t)(-dt) = - int_(a)^(0) f (-t) ft`
` = int_(0)^(a) f (-t) * dt `
`[:' int_(a)^(b) f (x) * dx = - int_(b)^(a) f(x) * dx]`
` = int_(0)^(a) f (-x) * dx `
` :. ` Equation (1) becomes
`int_(-a)^(a) f(x) * dx = int_(0)^(a) f (-x) * dx + int_(0)^(a) f (x) * dx `
` = int_(0)^(a) [f(-x)+f(x)]*dx ` ....(2)
Case I : If (x) is an even function, then ` f(-x) = f (x) `.
Thus , equation (2) becomes
`int_(-a)^(a) f (x) * dx = int_(0)^(a) [f(x)+f(x)] * dx = 2 int_(0)^(a) f(x) * dx `
Case II : If f (x) is an odd function, then `f(-x) = - f(x)`.
Thus equation (2) becomes,
`int_(-a)^(a) f(x) * dx = int_(0)^(a) [-f(x)+f(x)]*dx = 0`
Hence `int_(-a)^(a) f(x) * dx = {{:(2int_(0)^(a) f (x) * "dx,if f (x) is an even function "),(" 0, if f (x) is an odd function "):}`