Question

$\underset{-\mathrm{\pi }/4}{\overset{\mathrm{\pi }/4}{\int }}\left|\tan x\right|dx$

Answer 1

$$\begin{gathered} {\int }_{\frac{-\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{4}}\left|\tan x\right|dx \\ ={\int }_{\frac{-\mathrm{\pi }}{4}}^0-\tan xdx+{\int }_0^{\frac{\mathrm{\pi }}{4}}\tan xdx \\ ={\left[\log \left(\cos x\right)\right]}_{\frac{-\mathrm{\pi }}{4}}^0+{\left[-\log \left(\cos x\right)\right]}_0^{\frac{\mathrm{\pi }}{4}} \\ =-\log \frac{1}{\sqrt{2}}-\log \frac{1}{\sqrt{2}} \\ =2\log \sqrt{2} \\ =\log 2 \end{gathered}$$