A) \[\frac{1}{3}(4\pi +1)\]
B) \[\frac{4\pi }{3}-2\log \tan \frac{5\pi }{12}\]
C) \[\frac{4\pi }{3}+\log \tan \frac{5}{12}\]
D) none of these
Correct Answer: B
Solution :
Let\[f(x)=\frac{x\sin x}{x{{\cos }^{2}}x}=x\sec x\tan x\] \[\Rightarrow \]\[f(x)\]is an even function. \[\Rightarrow \] \[\int_{-\pi /3}^{\pi /3}{x\sec x\tan xdx}\] \[=2\int_{0}^{\pi /3}{x\sec x\tan xdx}\] \[=2\left[ x\sec x-\log \left( \tan \left( \frac{\pi }{4}+\frac{x}{2} \right) \right) \right]_{0}^{\pi /3}\] \[=\frac{4\pi }{3}-2\log \tan \frac{5\pi }{12}\]
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