Question

Solve $\int \frac{\sin 2x}{{\sin }^{4}x+{\cos }^{4}x}dx$.

Answer 1

$sin2x=2sinxcosx$

$sinx=\frac{tanx}{secx}$

$cosx=\frac{1}{secx}$

$\int \frac{2tanxse{c}^{2}xdx}{ta{n}^{4}x+1}$

Putting $tanx=m$

$se{c}^{2}xdx=dm$

$\int \frac{2mdm}{m^4+1}$

Putting $m^2=u$

$2mdm=du$

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

$ta{n}^{-1}u+C$

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