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