First slide

Methods of integration

Question

The value of $\int \frac{\sec x d x}{\sqrt{\sin (2 x + θ) + \sin θ}}$ is

Moderate

A

$\sqrt{(\tan x + \tan θ) \sec θ} + C$
B

$\sqrt{2 (\tan x + \tan θ) \sec θ} + C$
C

$\sqrt{2 (\sin x + \tan θ) \sec θ} + C$
D

none of these

Solution

The given integrals can be written as
$\begin{array}{l} \int \frac{\sec x d x}{\sqrt{2 \sin (x + θ) \cos x}} \\ = \frac{1}{\sqrt{2}} \int \frac{\sec^{3 / 2} x d x}{\sqrt{\sin x \cos θ + \sin θ \cos x}} \\ = \frac{1}{\sqrt{2}} \int \frac{\sec^{2} x d x}{\sqrt{\tan x \cos θ + \sin θ}} \\ = \frac{1}{\sqrt{2}} \int \frac{d t}{\sqrt{t \cos θ + \sin θ}} \\ = \frac{2}{\sqrt{2}} \sqrt{t \cos θ + \sin θ} + C \\ = \sqrt{2 (\tan x \sec θ + \tan θ \sec θ)} + C \end{array}$

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