Question

$\int \frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$ is equal to ${\tan }^{-1}\left(f\right(x{)}^n)+C$, then which of the following is/are correct ?

• A. $f\left(x\right)=\tan x$
• B. $f\left(x\right)=\cot x$
• C. $n=2$
• D. $n=4$

Answer 1

Answer: (A), (C)

$n=2$

$I=\int \frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$
$=\int \frac{2\sin x\cos x}{{\sin }^{4}x+{\cos }^{4}x}dx$
$=\int \frac{2\tan x{\sec }^{2}x}{1+{\tan }^{4}x}dx$
Let ${\tan }^{2}x=t\Rightarrow 2\tan x{\sec }^{2}xdx=dt$
$\Rightarrow I=\int \frac{dt}{1+t^2}={\tan }^{-1}t+C={\tan }^{-1}\left({\tan }^{2}x\right)+C$