Question

$\underset{0}{\overset{\mathrm{\pi }/2}{\int }}\log \tan xdx.$

Answer 1

$$\begin{gathered} \mathrm{Let},I={\int }_0^{\frac{\mathrm{\pi }}{2}}\log \tan xdx...\left(\mathrm{i}\right) \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\log \tan \left(\frac{\mathrm{\pi }}{2}-x\right)dx\left[\mathrm{Using},{\int }_0^{a}f\left(x\right)dx={\int }_0^{a}f\left(a-x\right)dx\right] \\ ={\int }_0^{\frac{\mathrm{\pi }}{2}}\log cotxdx...\left(\mathrm{ii}\right) \\ \mathrm{Adding}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)\mathrm{we}\mathrm{get} \\ 2I\mathit{=}{\mathit{\int }}_{\mathit{0}}^{\frac{\pi }{\mathit{2}}}\mathit{log}\mathit{}\mathit{tan}x\mathit{d}x\mathit{+}{\mathit{\int }}_{\mathit{0}}^{\frac{\pi }{\mathit{2}}}\mathit{log}\mathit{}cotx\mathit{}dx \\ \mathit{}\mathit{}\mathit{}={\mathit{\int }}_{\mathit{0}}^{\frac{\pi }{\mathit{2}}}\mathit{log}\left(\mathit{tan}x\mathit{\times }cotx\right)dx \\ \mathit{}\mathit{}\mathit{}={\mathit{\int }}_{\mathit{0}}^{\frac{\pi }{\mathit{2}}}\mathit{log}\mathit{1}\mathit{}dx\mathit{=}\mathit{0} \\ \mathrm{Hence},\mathit{}I\mathit{=}\mathit{0} \end{gathered}$$