First slide

Methods of integration

Question

$\int \frac{\sin 2 x}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} d x$

Easy

A

$\frac{1}{a^{2} + b^{2}} \log |a^{2} \sin^{2} x + b^{2} \cos^{2} x| + C$
B

$\frac{1}{a^{2} - b^{2}} \log |a^{2} \sin^{2} x + b^{2} \cos^{2} x| + C$
C

$\frac{1}{a^{2} - b^{2}} \log |a^{2} \sin^{2} x - b^{2} \cos^{2} x| + C$
D

$\frac{1}{a^{2} + b^{2}} \log |a^{2} \sin^{2} x - b^{2} \cos^{2} x| + C$

Solution

$\begin{array}{l} \frac{d}{d x} (a^{2} \sin^{2} x + b^{2} \cos^{2} x) = (a^{2} - b^{2}) \sin 2 x \\ Now, \int \frac{\sin 2 x}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} d x \\ = \frac{1}{(a^{2} - b^{2})} \int \frac{(a^{2} - b^{2}) \sin 2 x}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} d x \\ = \frac{1}{(a^{2} - b^{2})} \int \frac{\frac{d}{d x} (a^{2} \sin^{2} x + b^{2} \cos^{2} x)}{a^{2} \sin^{2} x + b^{2} \cos^{2} x} d x \\ = \frac{1}{(a^{2} - b^{2})} \log |a^{2} \sin^{2} x + b^{2} \cos^{2} x| + C \end{array}$

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