A) \[-\ln 2\]
B) \[\frac{{{\ln }^{2}}2}{2}\]
C) \[\frac{-{{\ln }^{2}}2}{2}\]
D) \[-{{\ln }^{2}}2\]
Correct Answer: C
Solution :
\[I=\int\limits_{0}^{\pi /3}{\underbrace{\tan x}_{II}\,.\underbrace{\ln \,(\cos x)}_{I}\,dx}\] \[=-\ln \,(\cos x).\ln \,(\cos x)]_{0}^{\pi /3}\] \[-\underbrace{\int\limits_{0}^{\pi /2}{\tan x\,\ln \,(\cos x)dx}}_{I}\] \[2I=\left[ \ln \,{{(\cos x)}^{2}} \right]_{\pi /30}^{0}=0\,-({{\ln }^{2}}2)=-{{\ln }^{2}}2\] \[\Rightarrow \,I=\frac{-{{\ln }^{2}}2}{2}\]
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