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If $\int \frac{2 x - \sqrt{\sin^{- 1} x}}{\sqrt{1 - x^{2}}} d x = C - 2 \sqrt{1 - x^{2}} -$ $\frac{2}{3} \sqrt{f (x)}$ then $f (x)$ is equal to

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a

sin−1⁡x

b

2sin−1⁡x

c

sin−1⁡x3

d

3sin−1⁡x3

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detailed solution

Correct option is C

The integrand is equal to 2x1−x2−sin−1⁡x1−x2. The first part is of form −g′(x)∣g(x) where g(x)=1−x2and other part is of the form h(x)h′(x),h(x)=sin−1⁡x Hence the required integral C−21−x2−23sin−1⁡x3/2. Thus f(x)=sin−1⁡x3

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