A) \[\pi \]
B) \[\pi /2\]
C) \[2\pi \]
D) \[4\pi \]
Correct Answer: A
Solution :
[a] \[I=\int\limits_{0}^{2\pi }{\frac{1}{1+{{\tan }^{4}}x}}dx\] \[\Rightarrow \,\,\,I=2\int\limits_{0}^{\pi }{\frac{dx}{1+{{\tan }^{4}}x}}\] \[\Rightarrow \,\,\,I=4\int\limits_{0}^{\pi /2}{\frac{dx}{1+{{\tan }^{4}}x}}\] ?.(1) or \[\Rightarrow \,\,\,I=4\int\limits_{0}^{\pi /2}{\frac{dx}{1+{{\cot }^{4}}x}}\] ?..(2) Adding (1) and (2), we get \[2I=4\int\limits_{0}^{\pi /2}{dx}=4\times \frac{\pi }{2}\] \[\Rightarrow \,\,\,\,I=\pi \]
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