Question

$\int \frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}$ is equal to:

• A. $2{\tan }^{-1}\left({\tan }^{2}x\right)+C$
• B. ${\tan }^{-1}\left(x{\tan }^{2}x\right)+C$
• C. ${\tan }^{-1}\left({\tan }^{2}x\right)+C$
• D. none of the above

Answer 1

Answer: (C)

${\tan }^{-1}\left({\tan }^{2}x\right)+C$

Let $I=\int \frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$

Dividing the numerator and denominator by ${\cos }^{4}x$

$I=\int \frac{2\tan x.{\sec }^{2}x}{{\tan }^{4}x+1}dx$

Let $\tan x=u$

$\left({\sec }^{2}x\right)dx=du$

$=\int \frac{2u}{1+u^4}du$

Let $u^2=z$

$2udu=dz$

$I=\int \frac{dz}{1+z^2}$

$\therefore I={\tan }^{-1}z+c$

$\therefore I={\tan }^{-1}{u}^2+c$

$\therefore I={\tan }^{-1}\left({\tan }^{2}x\right)+c$