Mathematics - Definite Integration Question with Solution | TestHub

Let $I={\int }_0^{\frac{\pi }{2}}\log (\tan x)dx$....(i) By applying property, $\begin{array}{rl}{\int }_a^{b}f(x)dx& ={\int }_a^{b}f(a+b-x)dx\\ I& ={\int }_0^{\frac{\pi }{2}}\log \left(\tan \left(\frac{\pi }{2}-x\right)\right)dx\\ I& ={\int }_0^{\frac{\pi }{2}}\log (\cot x)dx\dots (ii)\end{array}$ Adding Eqs. (i) and (ii), $\begin{array}{rl}& 2I={\int }_0^{\frac{\pi }{2}}[\log (\tan x)+\log (\cot x)]dx\\ & 2I={\int }_0^{\frac{\pi }{2}}\log 1dx=0\end{array}$