A) 0
B) 1
C) \[\frac{\pi }{2}\]
D) \[\frac{\pi }{4}\]
Correct Answer: D
Solution :
\[I=\int_{0}^{\pi /2}{\frac{dx}{1+{{\tan }^{3}}x}=\int_{0}^{\pi /2}{\frac{{{\cos }^{3}}x}{{{\sin }^{3}}x+\cos {{x}^{3}}}}}dx\] ?..(i) \[=\int_{0}^{\pi /2}{\frac{{{\sin }^{3}}x}{{{\cos }^{3}}x+{{\sin }^{3}}x}dx}\] .....(ii) Adding (i) and (ii), we get \[2I=\int_{0}^{\pi /2}{dx\Rightarrow I=\frac{\pi }{4}.}\]
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