Question

Evaluate: $\int \frac{sin2x}{si{n}^{4}x+co{s}^{4}x}dx$

Answer 1

Given

$\int \frac{sin2x}{si{n}^{4}x+co{s}^{4}x}$

$=\int \frac{2sinxcosx}{si{n}^{4}x+co{s}^{4}x}dx$

Multiply numerator and denominator by $se{c}^{4}x$

$=\int \frac{2tanxcoxx}{si{n}^{4}x+co{s}^{4}x}dx$

Let u= tan x and $du=se{c}^{2}xdx$

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

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

Let $s=u^2$ and $ds=2udu$

$=\int \frac{1}{s^2+1}ds$

the internal of $\frac{1}{s^2+1}$ is $ta{n}^{-1}s$

$=ta{n}^{-1}s+C$

$=ta{n}^{-1}{u}^2+C$

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