Answer
$${\text{The integral diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\tan \theta d\theta } \cr
& {\text{The integrand is not defined for }}\theta = \frac{\pi }{2},{\text{ then}} \cr
& \int_0^{\pi /2} {\tan \theta d\theta } = \mathop {\lim }\limits_{a \to \pi /{2^ - }} \int_0^a {\tan \theta d\theta } \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{a \to \pi /{2^ - }} \left[ {\ln \left| {\sec \theta } \right|} \right]_0^a \cr
& = \mathop {\lim }\limits_{a \to \pi /{2^ - }} \left[ {\ln \left| {\sec a} \right| - \ln \left| {\sec 0} \right|} \right] \cr
& = \mathop {\lim }\limits_{a \to \pi /{2^ - }} \left[ {\ln \left| {\sec a} \right| - 0} \right] \cr
& = \mathop {\lim }\limits_{a \to \pi /{2^ - }} \left[ {\ln \left| {\sec a} \right|} \right] \cr
& {\text{Evaluating the limit}} \cr
& = \ln \left| {\sec \frac{\pi }{2}} \right| \cr
& = \infty \cr
& {\text{The integral diverges}} \cr} $$
