A) \[0\]
B) \[1\]
C) \[\pi /4\]
D) \[\pi /2\]
Correct Answer: D
Solution :
\[I=\int\limits_{\pi /4}^{\pi /4}{{}}(\underset{\begin{smallmatrix} \downarrow \\ odd\,\,f \end{smallmatrix}}{\mathop{x|x|}}\,+\underset{\begin{smallmatrix} \downarrow \\ odd\,\,f \end{smallmatrix}}{\mathop{{{\sin }^{3}}x}}\,+x\underset{\begin{smallmatrix} \downarrow \\ odd\,\,f \end{smallmatrix}}{\mathop{{{\tan }^{2}}x}}\,+1)dx\] \[I=\int\limits_{-\pi /4}^{\pi /4}{dx}=\frac{\pi }{2}\] \[[\because \,\,\int\limits_{-a}^{a}{f(x)}dx=0,\] as\[f(x)\]is an odd function]
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