Question

evaluate the integral of x e^(-x^2) from minus infinity to infinity. I already found the integral but I don't understand why the limits of this will give 0

Answer 1

well, e^(-x^2) is an even function

and x is an odd function

even times odd = odd

Answer 2

the integral from 0 to infinity is the opposite of it from 0 to -inf

Answer 3

@cpham did you try drawing it out... ?
The key is the symmetry about \(y=x\)... observe:$$f(x)=xe^{-x^2}\\f(-x)=(-x)e^{-(-x)^2}=-xe^{-x^2}=-f(x)$$