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