Just do (a) directly with polar co-ordinates.
That's how you prove the normal density integrates to one.
You can work this out from basics as suggested by MathEagle, but if you are allowed to assume some basic knowledge of the Normal distribution there is a shortcut, which I think is probably what you are intended to do--
After you make the substitution, you should have
which is equal to
Based on your knowledge of the Normal distribution, you should know the value of .
Take it from there.