I saw an integral like the following in a book: For $p, q$ nonnegative integers, $$\int_{e}^{\infty}\frac{r^{2p-2q-1}}{(\log r)^{2q+2}}dr$$ is finite IF and ONLY IF $q \geq p$. I am not sure how to prove it.
If I make the simple substitution $\log r=x$ then the integral above changes to $$\int_{1}^{\infty} e^{{(2p-2q)}x}x^{-(2q+2)}dx.$$ But even then I am not sure that I can see it is finite IF and ONLY IF $q \geq p$. Can anyone please tell me how to see both directions? It kind of makes sense in a graphical way but I wanted a rigorous proof.