Convergence of $\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$ Does this integral converge?
$$\int_3^{\infty} \frac{1}{(\ln(x))^2(x-\ln(x))}$$
I've been trying to solve this for the past 2 hours...literally. I know the answer is fairly simple, but I just can't think of it

Yes, this converges.  The integrand is bounded above by $2/(x (\log x)^2)$.  Then, substitute $y=\log x$ to get the convergent integral $\int_{\log 3}^\infty 2 y^{-2} dy$.
