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Question: The value of the integral \(\int_{- 1}^{1}{\frac{d}{dx}\left( \tan^{- 1}\frac{1}{x} \right)}dx\) is...

The value of the integral ∫−11ddx(tan⁡−11x)dx\int_{- 1}^{1}{\frac{d}{dx}\left( \tan^{- 1}\frac{1}{x} \right)}dx is

A

π2\frac{\pi}{2}

B

π4\frac{\pi}{4}

C

−π2- \frac{\pi}{2}

D

None of these

Answer

−π2- \frac{\pi}{2}

Explanation

Solution

∫−11ddx(tan⁡−11x)dx=−2[tan⁡−1(x)]01=−π2\int_{- 1}^{1}\frac{d}{dx}\left( \tan^{- 1}\frac{1}{x} \right)dx = - 2\lbrack\tan^{- 1}(x)\rbrack_{0}^{1} = - \frac{\pi}{2}.