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$\int \frac{{(\cot \sqrt{\tan^{- 1} x})}^{4} \cos e c^{2} \sqrt{\tan^{- 1} x} d x}{(1 + x^{2}) \sqrt{\tan^{- 1} x}} =$

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5cot5cot−1x+C

−5cot5cot−1x+C

15cot5tan−1x+C

−15cot5tan−1x+C

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detailed solution

Correct option is D

T=∫cottan−1x4cosec2tan−1xdx1+x2tan−1x Put tan−1⁡x=yT=∫coty4cosec2ydx Put cot⁡y=zcos⁡ec2ydy=−dzT=∫z4(−dz)=−z55+C=−15(cot⁡y)5+C=−15cot5⁡tanh−1x+C

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∫cottan−1x4cosec2tan−1xdx1+x2tan−1x=

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$\int \frac{{(\cot \sqrt{\tan^{- 1} x})}^{4} \cos e c^{2} \sqrt{\tan^{- 1} x} d x}{(1 + x^{2}) \sqrt{\tan^{- 1} x}} =$