$Evaluate \int \frac{\sin x}{\sin 3 x} d x$

Moderate

A

$\frac{1}{2 \sqrt{3}} \log |\frac{\sqrt{3} - \tan x}{\sqrt{3} + \tan x}| + C$
B

$\frac{1}{\sqrt{3}} \log |\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}| + C$
C

$\frac{1}{2 \sqrt{3}} \log |\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}| + C$
D

$\frac{1}{2} \log |\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}| + C$

Solution

$\begin{array}{l} I = \int \frac{\sin x}{\sin 3 x} d x = \int \frac{\sin x}{3 \sin x - 4 \sin^{3} x} d x \\ = \int \frac{1}{3 - 4 \sin^{2} x} d x \\ = \int \frac{\sec^{2} x}{3 \sec^{2} x - 4 \tan^{2} x} d x D i v i d i n g N r a n d D r b y \cos^{2} x \\ P u t t i n g \tan x = t a n d s e c^{2} x d x = d t, w e g e t \\ \begin{array}{l} I = \int \frac{d t}{3 (1 + t^{2}) - 4 t^{2}} = \int \frac{d t}{3 - t^{2}} = \int \frac{1}{(\sqrt{3})^{2} - t^{2}} d t \\ = \frac{1}{2 \sqrt{3}} \log |\frac{\sqrt{3} + t}{\sqrt{3} - t}| + C = \frac{1}{2 \sqrt{3}} \log |\frac{\sqrt{3} + \tan x}{\sqrt{3} - \tan x}| + C \end{array} \end{array}$

Get Instant Solutions

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

Recieve an sms with download link

SCAN ME