How do I perform this integration?

Question

I tried doing it by parts, but I keep getting stuck.

Show that $\int t\phi(t)dt=-\phi(t)+c$,

where $c$ is a constant

$\:\qquad\phi$ is the pdf of a standard normal variable.

Answer 1

$te^{-t^{2}/2}$ is the derivative of $-e^{-t^{2}/2}$. Hence its integral is $-e^{-t^{2}/2}+C$.