I was trying to solve
$\int_0^\infty x \exp \left( { - \frac{{{x^2}}}{2}} \right)\;{\text{d}}x$
I was trying to use $(f(g(x)))'=f'(g(x))\cdot g'(x)$
(So $[e^{f(x)}]'=[e^{f(x)}]\cdot f'(x)$ ?), but I got the wrong answer. (I substituted $0$ and $\infty$ to $-\exp \left(-\frac{x^2}{2}\right)$ and got 1. But the test solution says $\sqrt{e}$.)
Could you point out what's wrong with my solution?