I am asked to show that

for p > 0. I have a lot of it so far, and am basically on the cusp.

. Now the obvious calculus thing to do is integrate by parts and I'm done. But to do that, assuming I'm treating this like a Riemann integral, I have to evaluate

from 0 to 1. Well okay, 1 is easy. Log of 0 isn't defined, though, and I'm not sure how to do a proofy demonstration that this is whole thing is zero. So that leads me to think that maybe I shouldn't be treating this like a Riemann integral and instead should find some sequence of step functions which approaches what I'm trying to integrate, i.e. to argue it as a Lebesgue integral. But finding a sequence of step functions which approaches

seems daunting. Ideas?