First slide

Evaluation of definite integrals

Question

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

Moderate

A

$3 \sqrt{2} + π$
B

$2 \sqrt{3}$
C

$3 \sqrt{2} - π$
D

$2 \sqrt{3} + π$

Solution

$\begin{array}{l} Consider the integral I = \int_{0}^{\frac{1}{2}} \frac{x^{2}}{{(1 - x^{2})}^{\frac{3}{2}}} d x \\ Substitute x = \sin θ, d x = \cos θ d θ and the limits are 0, \frac{π}{6} \\ Hence, the given integral becomes \end{array}$
$\begin{matrix} I = \int_{0}^{\frac{π}{6}} \frac{\sin^{2} θ}{\cos^{3} θ} \cos θ d θ \\ = \int_{0}^{\frac{π}{6}} \tan^{2} θ d θ \\ = \int_{0}^{\frac{π}{6}} \sec^{2} θ - 1 d θ \\ = {[\tan θ - θ]}_{0}^{\frac{π}{6}} \\ = \frac{1}{\sqrt{3}} - \frac{π}{6} \end{matrix}$
$Therefore, \frac{k}{6} +π = \frac{1}{\sqrt{3}} - \frac{π}{6} = \frac{6 - \sqrt{3} π}{6 \sqrt{3}} + π, it implies k = 2 \sqrt{3}$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App