On part (a), did you try directly computing both sides and solving the resulting system of equations for ?
You're right. Your single equation doesn't prove part (a) yet. Let me just review the situation for you:
We want to compute , where , for . Taking the integral of the left hand side and evaluating the right hand side gives
Expanding the terms:
Now, we have two systems of equations (by comparing the coefficients of each and on both sides):
Solving the system shows that .
For part (b), I might apply the same strategy and see what happens.