Answer
Diverges.
Work Step by Step
We have: $\int_{1}^{2} \dfrac{1}{x \ln x} d x= \lim\limits _{a \to 1^{+}}\int_{a}^{2} \dfrac{1}{x \ln x} d x $
or, $= \lim\limits _{a \to 1^{+}} \ln [\ln x]|_a^2$
or, $=\lim\limits _{a \to 1^{+}} (\ln (\ln 2)-\ln (\ln a))$
or, $=\infty$
Thus, the limit does not exist, and so, the integral diverges.