I'm trying to compute . I know the answer is , but not sure how to prove this. can anyone help? thank you.
Apart from the fact that what you wrote doesn't seem to be true in the general case (though a simmilar asymptotic behaviour here happens, if I understood correctly what you probably meant to say), you have some serious mistakes here: the derivative of is is its antiderivative or primitive function.
Perhaps you tried to come up with a rather sui generis "proof" in the spirit of the integral test, but nevertheless it isn't quite so, and some conditions must be fulfilled in the general case.