Question

The value of $\int \frac{sin2x}{si{n}^{4}x+co{s}^{4}x}dx$ is

• A. $ta{n}^{-1}\left(co{t}^{2}x\right)+C$
• B. $-ta{n}^{-1}\left(cot2x\right)+C$
• C. $ta{n}^{-1}\left(sin2x\right)+C$
• D. $ta{n}^{-1}\left(ta{n}^{2}x\right)+C$

Answer 1

Answer: (B)

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