Determining what values a function converges at

[nigba]

Homework Statement

For what values of p and q does the integral

dx / (x^p * (ln(x))^q) from 1 to infinity

converge?

Homework Equations

integration

The Attempt at a Solution

I have no idea how to start figuring this out. I've tried trig substitutions but can't find something that actually makes progress.

 

[ystael]

You don't need to actually evaluate the integral of (find a primitive of) $$x^{-p} \log^{-q} x$$ for generic $$p$$ and $$q$$. Instead, compare the growth behavior of this function to simpler functions whose integrals you know converge or diverge, and try to find critical values of $$p$$ and $$q$$ where the behavior changes.