I often find my self lost and doing the same thing when solving math problems. My professor seems to know exactly what will work and what will not in an integration problem, how do I do that?

For example, consider this:
 $e^{-x^2}$

when I see this i would look at the integration methods I know, pick the ones I can apply, and try them all (Very time consuming). So I would try u substitution first

But there are $3$ different possible choices for u!
$u = x^2$
or 
$u = x$
or 
$u = e^{-x^2}$

I would try these, then move on to the by parts method, trig substitution, partial fractions, and so on. Do you see where my problem is? It's making integration so frustrating to do!

sure, it is sometimes obvious what Not to use (so partial fractions here is a no no), but still I end up with so many possible choices, and every choice branches off into even more choices!

Sometimes I try to skip some methods, only to find that they were the solution!

How do I predict what will work and what will not?