|
Note that what is meant by $\displaystyle \int\limits_0^{\frac{\pi}{2}} \tan x \, dx$ is actually $\displaystyle \lim_{t \to \frac{\pi}{2}^-} \int\limits_0^t \tan x \, dx$ since $\tan x$ is defined on $[0, \frac{\pi}{2})$ but not at $\frac{\pi}{2}$
$\begin{array}{ll} \displaystyle \int\limits_0^{\frac{\pi}{2}} \tan x \, dx & \displaystyle = \lim_{t \to \frac{\pi}{2}^-} \int\limits_0^t \tan x \, dx \\ & \\ & \displaystyle = \lim_{t \to \frac{\pi}{2}^-} \ln |\sec x| \bigg|_0^t \\ & \\ & \displaystyle = \lim_{t \to \frac{\pi}{2}^-} \ln |\sec t| - \ln |\sec 0| \\ & \\ & \displaystyle = \lim_{t \to \frac{\pi}{2}^-} \ln |\sec t| \\ & \\ & \displaystyle = \infty. \end{array}$
| 
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)