text: [help~!!!][calculus - integrals]how can i prove this? suppose  f:[-a, a] --> r  is either an even or an odd function. show that  s(from 0 to x) f(s)ds  is therefore either an odd or an even function, respectively.  *i coudn't place integral symbol so i put 's' instead. and 0 would be subscript (at below) and x would be superscript (at above) of integral 's'   thank you!!! suppose that f(x) is an even function, then f(-x)=f(x)  let g(x) = int^x_0 f(x) dx,  then g(-x) = int^(-x)_0 f(s) ds  let s=-u so ds = -du then  g(-x) =  - int^x_0 f(-u) du        = - int^x_0 f(u) du        = -g(x)  the function is odd.  now suppose that f(x) is odd, f(-x)=-f(x) then let g(x) = in^x_0 f(s) ds and then g(-x) = int^-x_0 f(s) ds  let s=-u then ds=-du so  g(-x) = -int^x_0 f(-u) du        = -int^x_0 -f(u) du        = int^x_0 f(u) du        = g(x)  so the function is even.  if you just look at a graph the result is counter-intuitive, but you have to remember that the bounds of integration matter!