A) \[2\pi \]
B) \[\pi \]
C) \[\pi /2\]
D) \[\pi /4\]
Correct Answer: A
Solution :
\[I=\int_{-1}^{3}{\left[ {{\tan }^{-1}}\left( \frac{x}{{{x}^{2}}+1} \right){{\tan }^{-1}}\left( \frac{{{x}^{2}}+1}{x} \right) \right]}\,dx\] \[=\int_{-1}^{3}{{{\tan }^{-1}}\left| \frac{\frac{x}{{{x}^{2}}+1}+\frac{{{x}^{2}}+1}{x}}{1-\frac{x}{{{x}^{2}}+1}.\frac{{{x}^{2}}+1}{x}} \right|}\,dx\] \[=\int_{-1}^{3}{{{\tan }^{-1}}\infty \,dx=\frac{\pi }{2}\int_{-1}^{3}{dx=\frac{\pi }{2}\left[ x \right]_{1}^{3}}}\] \[=2\pi \]
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