A) \[\frac{\pi }{4}\]
B) \[\frac{\pi }{2}\]
C) 0
D) None of these
Correct Answer: C
Solution :
\[I=\int_{0}^{\pi /2}{\log \tan xdx}\] ?..(i) \[\therefore \] \[I=\int_{0}^{\pi /2}{\log \tan \left( \frac{\pi }{2}-x \right)}dx\] \[=\int_{0}^{\pi /2}{\log \cot x\,}dx\] ?.(ii) On adding Eqs. (i) and (ii), \[2I=\int_{0}^{\pi /2}{(\log \tan x+\log \cot x)\,}dx\] \[=\int_{0}^{\pi /2}{\log (\tan x.\cot x)\,}dx\] \[=\int_{0}^{\pi /2}{\log 1}\,dx\] \[\Rightarrow \] \[2I=0\] \[\Rightarrow \] \[I=0\]
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