how to integrate x(1/sqrt(2pi))(exp(-(x-y)^2/2)) with respect to x from infinity to infinity? the answer is y. but how do you get that?
This integral is the mean of X. When $\displaystyle X\sim N(Y,1)$.
You can let $\displaystyle Z=X-Y\sim N(0,1)$.
There are many ways to show that $\displaystyle \int_{\infty}^{\infty}z e^{-z^2/2}dz=0$.
The easiest way is to note that the integrand is an odd function.