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