Answer
The integral diverges.
Work Step by Step
We have
$$
\int_{0}^{\pi/2} \cot \theta d\theta=\int_{0}^{\pi/2} \frac{\cos \theta}{\sin\theta} d\theta=\int_{0}^{\pi/2} \frac{d\sin\theta}{\sin\theta} \\
=\ln(\sin\theta)|_0^{\pi/2}=\ln 1- \ln 0=-\infty.
$$
Hence, the integral diverges.
