Answer
$${\text{diverges}}$$
Work Step by Step
$$\eqalign{
& \int_0^{\pi /2} {\tan x} dx \cr
& {\text{The integrand is undefined for }}x = \pi /2,{\text{ so the integral can be represented as}} \cr
& \int_0^{\pi /2} {\tan x} dx = \mathop {\lim }\limits_{k \to \pi /{2^ - }} \int_0^k {\tan x} dx \cr
& {\text{Integrate}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to \pi /{2^ - }} \left[ {\ln \left| {\sec x} \right|} \right]_0^k \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to \pi /{2^ - }} \left[ {\ln \left| {\sec k} \right| - \ln \left| {\sec 0} \right|} \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\lim }\limits_{k \to \pi /{2^ - }} \left[ {\ln \left| {\sec k} \right|} \right] \cr
& \,{\text{Calculate the limit when }}k \to \pi /{2^ - } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \ln \left| {\sec {{\left( {\frac{\pi }{2}} \right)}^ - }} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \ln \left| {\frac{1}{{\cos {{\left( {\pi /2} \right)}^ - }}}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \ln \left| {\frac{1}{{{0^ - }}}} \right| \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \ln \left( \infty \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = + \infty \cr
& {\text{Then}}{\text{,}} \cr
& \int_0^{\pi /2} {\tan x} dx = + \infty \cr
& {\text{The integral diverges}} \cr} $$
