Answer
Diverges
Work Step by Step
We have: $\int_{0}^{\pi / 2} \cot \theta d \theta=\lim\limits_{b \to 0^{+}}\int_{0}^{\pi / 2} \cot \theta d \theta\\
=\lim\limits_{b \to 0^{+}} [\ln |\sin \theta|]_{b}^{\pi / 2}\\
=\lim\limits_{b \to 0^{+}}[\ln 1-\ln |\sin b|]\\
=-\lim\limits_{b \to 0^{+}}[\ln |\sin b|]\\
=+\infty$
Thus, the limit does not exist, and so, the integral diverges.
