A) zero
B) 2
C) \[\pi /3\]
D) \[\pi /4\]
Correct Answer: A
Solution :
\[I=\int_{0}^{\pi /2}{\log (\tan x)dx}\] ?(i) \[I=\int_{0}^{\pi /2}{\log \tan \left( \frac{\pi }{2}-\pi \right)}\,dx\] \[I=\int_{0}^{\pi /2}{\log \cot x\,dx}\] ?(ii) Adding (i) and (ii) \[2I=\int_{0}^{\pi /2}{[\log (\tan x)+\log (\cot x)]dx}\] \[2I=\int_{0}^{\pi /2}{\log (\tan x\cot x)\,dx}\] \[2I=\int_{0}^{\pi /2}{\log 1.dx\Rightarrow 2I=0\Rightarrow I=0}\]
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