Answer
Diverges
Work Step by Step
Since
\begin{align*}
\int_{0}^{\pi / 2} \tan \theta d \theta& =\lim _{b \rightarrow\left(\frac{\pi}{2}\right)^{-}}\int_{0}^{b} \tan \theta d \theta \\
&=\lim _{b \rightarrow\left(\frac{\pi}{2}\right)^{-}}[-\ln |\cos \theta|]_{0}^{b}\\
&=\lim _{b \rightarrow\left(\frac{\pi}{2}\right)^{-}}[-\ln |\cos b|+\ln 1]\\
&=\lim _{b \rightarrow\left(\frac{\pi}{2}\right)^{-}}[-\ln |\cos b|]\\
&=+\infty
\end{align*}
Then the integral diverges.
