If the value of the integral $\int_{0}^{1 / 2} \frac{x^{2}}{\left(1-x^{2}\right)^{3 / 2}} d x$ is $\frac{k}{6}$, then $k$ is equal to:

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Indefinite Integrals

If the value of the integral

\int_{0}^{1/2} \frac{x ^{2}}{( 1 - x ^{2} ) ^{3/2}} d x

\frac{k}{6}

, then

k

is equal to:

Answer: (C)

Substitute

x = sin θ

d x = cos θ d θ

, with the limits running from

θ = 0

θ = π /6

. The integrand becomes

\frac{sin ^{2} θ}{cos ^{3} θ} cos θ = tan^{2} θ

, so the integral is

\int_{0}^{π /6} (sec^{2} θ - 1) d θ = [tan θ - θ]_{0}^{π /6} = \frac{1}{3} - \frac{π}{6} = \frac{2 3 - π}{6}

. Hence

k = 23 - π

JEE Main · 2020 · 03 Sep 2020 · Shift 2 · Single correct option

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