lf the value of the integral ∫01/2 x21−x23/2dx is k6, then k is equal to

# lf the value of the integral ${\int }_{0}^{1/2} \frac{{x}^{2}}{{\left(1-{x}^{2}\right)}^{3/2}}dx$ is $\frac{k}{6}$, then $k$ is equal to

1. A

$2\sqrt{3}+\pi$

2. B

$2\sqrt{3}-\pi$

3. C

$3\sqrt{2}+\pi$

4. D

$3\sqrt{2}-\pi$

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### Solution:

We have,

${\int }_{0}^{1/2} \frac{{x}^{2}}{{\left(1-{x}^{2}\right)}^{3/2}}dx=\frac{k}{6}$

$⇒{\int }_{0}^{\pi /6} \frac{{\mathrm{sin}}^{2}\theta }{{\mathrm{cos}}^{3}\theta }\mathrm{cos}\theta d\theta =\frac{k}{6}$, where $x=\mathrm{sin}\theta$

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