Solution:
\( \displaylines{\int_2^{\infty}\frac{1}{x\ln\left(x\right)^2}\differentialD x=\int_{\ln\left(2\right)}^{\infty}\frac{1}{u^2}du\\ =-\frac{1}{u}\left|^{\infty}\right._{\ln\left(2\right)}\\ =\frac{1}{\ln\left(2\right)}\\ \text{Convergent by an integral comparison test}} \)