convergence of an improper interval with two functions that are battling each other to see if the integral of the product converges

Question

Boiled down, I have two functions of x and I need to know if the indefinite integral of their product converges. In particular, if:

$$y=x\exp{-x^2}$$

then how can I show (if its true) that the indefinite integral, from minus to plus infinity converges to some value?

Answer 1

First you should note that $$\int_a^bxe^{-x^2}dx=-\frac{1}{2}e^{-x^2}|_a^b.$$