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The value of $\int_{0}^{\sin^{2} x} \sin^{- 1} \sqrt{t} dt + \int_{0}^{\cos^{2} x} \cos^{- 1} \sqrt{t} dt$ is

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a

π2

b

1

c

π4

d

None of these

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detailed solution

Correct option is C

f(x)=∫0sin2⁡x sin−1⁡tdt+∫0cos2⁡x cos−1⁡tdtf′(x)=sin−1⁡sin2⁡x2sin⁡xcos⁡x+cos−1⁡cos2⁡x(−2cos⁡xsin⁡x)=sin⁡2xsin−1⁡(sin⁡x)−cos−1⁡(cos⁡x)=sin⁡2x(x−x)=0f (x) = constantFind f (x) for any value of x∫01/2 π2dx=π2⋅12=π4

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The value of $\int_{0}^{1} [\prod_{r = 1}^{n} (x + r)] (\sum_{k = 1}^{n} \frac{1}{x + k}) dx$ is

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