integrating $tanx$ over $[0,\frac{\pi }{2}]$

Note that what is meant by ${\displaystyle \underset{0}{\overset{\frac{\pi }{2}}{\int }}}\tan xdx$ is actually $\underset{t\to {\frac{\pi }{2}}^{-}}{lim}{\displaystyle \underset{0}{\overset{t}{\int }}}\tan xdx$, since $\tan x$ is defined on $[0,\frac{\pi }{2})$ but not at $\frac{\pi }{2}$.