Determine whether $\int_{0}^{\pi /2}\tan x\,dx$ converges or diverges.

Solution The function $f (x)$ = $\tan x$ is continuous on $\left[ 0,\,\dfrac{\pi }{2}\right)$ but is not defined at $\dfrac{\pi }{2}$, so $\int_{0}^{\pi /2}\tan x\,dx$ is an improper integral. $$\begin{eqnarray*} \int_{0}^{\pi /2}\tan x~dx\hbox{:} \lim\limits_{t\,\rightarrow \left( \pi /2\right) ^{-}}\int_{0}^{t}\tan x\,dx&=&\lim\limits_{t\,\rightarrow \ (\pi /2) ^{-}}\big[ \ln \vert \sec x \vert\big] _{0}^{t}\\[4pt] &=&\lim\limits_{t\,\rightarrow (\pi /2) ^{-}}[\ln \vert \sec t \vert -\ln \vert \sec 0 \vert ] \\[4pt] &=& \lim\limits_{t\,\rightarrow (\pi /2) ^{-}}\ln (\sec t)=\infty \end{eqnarray*}$$

So, $\int_{0}^{\pi /2}\tan x\,dx$ diverges.