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Question: The value of \(\int _ { - 1 } ^ { 3 } \tan ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1 } \right) +...

The value of ∫−13tan⁡−1(xx2+1)+tan⁡−1(x2+1x)dx\int _ { - 1 } ^ { 3 } \tan ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1 } \right) + \tan ^ { - 1 } \left( \frac { x ^ { 2 } + 1 } { x } \right) d x is

A

2π2 \pi

B

π\pi

C

π2\frac { \pi } { 2 }

D

π4\frac { \pi } { 4 }

Answer

2π2 \pi

Explanation

Solution

I=∫−13[tan⁡−1(xx2+1)+cot⁡−1(xx2+1)]dxI = \int _ { - 1 } ^ { 3 } \left[ \tan ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1 } \right) + \cot ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1 } \right) \right] d x

(∵tan⁡−1(x)+cot⁡−1(x)=π2)\left( \because \tan ^ { - 1 } ( x ) + \cot ^ { - 1 } ( x ) = \frac { \pi } { 2 } \right).