$\int\limits_0^{\pi /4} {\tan x} dx$

: $\int\limits_0^{\pi /4} {\tan x} dx = \left[ {\log \sec x} \right]_0^{\pi /4}$

$= \log \left( {\sec \cfrac{\pi }{4}} \right) - \log \left( {\sec \theta } \right) = \log \sqrt 2 - \log 1 = \cfrac{1}{2}\log 2$