But I don't think you can just yank the out of the integral and divide both sides by it. In addition, if you differentiate the RHS, you don't get the integrand on the LHS.
The integral you're trying to do is not trivial. Mathematica gives hypergeometric functions thus:
Now I thought that I could simply do this
which would mean that
I realized that this is dead wrong as soon as I thought of setting a to zero, but I don't know where my mistake is. Could someone please explain why this doesn't work?