∫ Sin 2 X Sin 4 X + Cos 4 X D X - Mathematics

Sum
$\int\frac{\sin 2x}{\sin^4 x + \cos^4 x} \text{ dx }$

Solution

$\text{ Let I } = \int\frac{\sin 2x}{\sin^4 x + \cos^4 x}dx$
$= \int\frac{2 \text{ sin x }\cdot \text{ cos x dx}}{\sin^4 x + \cos^4 x}$
$\text{Dividing numerator and denominator by} \cos^4 x$
$\Rightarrow \int\frac{2 \frac{\text{ sin x }\cdot \text{ cos x}}{\cos^4 x}dx}{1 + \tan^4 x}$
$\Rightarrow \int\frac{2 \tan x \cdot \text{ sec}^2 x dx}{1 + \left( \tan^2 x \right)^2}$
$\text{ Putting tan}^2 x = t$
$\Rightarrow 2 \tan x \cdot \text{ sec}^2 \text{ x dx}$
$\therefore I = \int\frac{dt}{1 + t^2}$
$= \tan^{- 1} t + C$
$= \tan^{- 1} \left( \text{ tan}^2 x \right) + C......... \left[ \because t = \tan {}^2 x \right]$

Concept: Indefinite Integral Problems
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RD Sharma Class 12 Maths
Chapter 19 Indefinite Integrals
Revision Excercise | Q 41 | Page 203