Answer
Diverges
Work Step by Step
Use a direct comparison test.
We can see that $0 \leq \dfrac{1}{x} \leq \dfrac{1}{\ln x}$ for all $x \gt 2$
Now, $\int_{2}^{\infty} \dfrac{dx}{\ln x}=\lim\limits_{a \to \infty}\int_{2}^{a} \dfrac{dx}{\ln x} \\=\infty$
and the integral $\int_{2}^{a} \dfrac{dx}{\ln x}$ diverges; this implies that $\int_{2}^{\infty} \dfrac{dx}{\ln x}$ diverges by the direct comparison test.