First slide

Evaluation of definite integrals

Question

The value of the integral $\int_{0}^{π / 2} \frac{d x}{1 + \frac{1}{6} \sin^{2} x}$ is

Easy

A

$\frac{π}{2} \sqrt{\frac{6}{7}}$
B

$\frac{π}{3} \sqrt{\frac{2}{3}}$
C

$\frac{π}{6}$
D

$\frac{π}{2}$

Solution

$I = \int_{0}^{π / 2} \frac{d x}{1 + \frac{1}{6} \sin^{2} x} = \int_{0}^{π / 2} \frac{\sec^{2} x d x}{\sec^{2} x + \frac{1}{6} \tan^{2} x}$
$\begin{array}{l} = \frac{6}{7} \int_{0}^{\infty} \frac{d t}{6 / 7 + t^{2}} (t = \tan x) \\ = \frac{6}{7} \times {\sqrt{\frac{7}{6}} \tan^{- 1} t \sqrt{\frac{7}{6}}|}_{0}^{\infty} = \sqrt{\frac{6}{7}} \cdot \frac{π}{2} . \end{array}$

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