Answer
Divergent
Work Step by Step
Let \[I=\int_{2}^{\infty}\frac{dx}{x\ln x}\]
This is improper integral
\[\Rightarrow I=\lim_{t\rightarrow \infty}\int_{2}^{t}\frac{dx}{x\ln x} \;\;\;\ldots (1)\]
\[I=\lim_{t\rightarrow \infty}\left[\ln|\ln x|\right]_{2}^{t}\]
\[I=\lim_{t\rightarrow \infty}\left[\ln|\ln t|-\ln|\ln 2|\right]\]
$\;\;\;\;\;\;\;\;\;\;\;\;\;\Rightarrow$does not exist
Since limit on R.H.S. of (1) does not exist so $I$ is divergent.
