∫ 1 Cos 2 X + 3 Sin 2 X D X - Mathematics

Sum
$\int\frac{1}{\cos 2x + 3 \sin^2 x} dx$

Solution

$\text{ Let I }= \int \frac{1}{\text{ cos } \left( \text{ 2x }\right) + 3 \sin^2 x}\text{ dx }$
$= \int \frac{1}{\left( 1 - 2 \sin^2 x \right) + 3 \sin^2 x}\text{ dx }$
$= \int \frac{1}{1 + \sin^2 x}\text{ dx }$
$\text{Dividing numerator and denominator by} \cos^2 x$
$\Rightarrow I = \int\frac{\sec^2 x}{\sec^2 x + \tan^2 x}dx$
$= \int\frac{\sec^2 x}{1 + \tan^2 x + \tan^2 x}dx$
$= \int \frac{\sec^2 x}{1 + 2 \tan^2}dx$
$= \int \frac{\sec^2 x}{1 + \left( \sqrt{2} \tan x \right)^2}dx$
$\text{ Let }\sqrt{2} \tan x = t$
$\Rightarrow \sqrt{2} \sec^2 x \text{ dx }= dt$
$\Rightarrow \sec^2 x \text{ dx } = \frac{dt}{\sqrt{2}}$
$\therefore I = \frac{1}{\sqrt{2}} \int \frac{dt}{1 + t^2}$
$= \frac{1}{\sqrt{2}} \tan^{- 1} \left( t \right) + C$
$= \frac{1}{\sqrt{2}} \tan^{- 1} \left( \sqrt{2} \tan x \right) + C$

Concept: Indefinite Integral Problems
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Exercise 19.22 | Q 11 | Page 114