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If $I_{n} = \int_{0}^{1} {(\cos^{- 1} x)}^{n} d x$ then $I_{6} - 360 I_{2}$ is given by

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a

6π25−24π23

b

6π25−120π23

c

6π25

d

6π25−4π23

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detailed solution

Correct option is B

Integrating by parts, we obtainIn=∫01 cos−1⁡ xndx=xcos−1⁡ x01+∫01 ncos−1⁡ xn−1x1−x2dx=n∫01 x1−x2cos−1⁡ xn−1dx=n−1−x2cos−1⁡ xn-101−∫01 (n−1)cos-1⁡ xn-2dx=nπ2n−1−n(n−1)In−2I6=6⋅π25−6.5I4=6⋅π25−304π23−12I2I6−360I2=6⋅π25−120π23

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Let $I_{n} = \int_{0}^{π / 2} x^{n} \cos x d x$ then $I_{8} + 56 I_{6}$ is equal to

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