I didn't integrate the expected value. That's how the expected value is defined for continuous distributions (like the normal):
, where is the pdf of the distribution.
Here's a site that evaluated the integral the way I did: The Normal Distribution
Note that
Thus, .
is a lot of work.
The integrand is an odd function so the integral must be zero.
I just commented on that in my review of that calc book.
Noticing that a function is even or odd can save some work.
As for
I first show that the normal is a valid density with that polar coordinate trick (multiplying it by itself)
and then we have
since this is a valid density of a N(0,1).