<s>[INST] Evaluate $\int_0^{\frac{\pi}{2}}\frac{\sin x\cos x}{\sin^4x+\cos^4x}dx$ 
Evaluate
  $$
\int_0^{\frac{\pi}{2}}\frac{\sin(x)\cos(x)}{\sin^4(x)+\cos^4(x)}dx
$$

I used the substitution $\sin x =t$, then I got the integral as $$\int_0^1 \frac{t}{2t^4-2t^2+1}dt $$ 
After that I don't know how to proceed. Please help me with this.
 [/INST] Perform the change of variable $y=\sin^2 x$,
$\begin{align}J&=\int_0^{\frac{\pi}{2}}\frac{\sin(x)\cos(x)}{\sin^4(x)+\cos^4(x)}dx\\
&=\frac{1}{2}\int_0^1 \frac{1}{x^2+(1-x)^2}\,dx\\
&=\frac{1}{2}\int_0^1 \frac{1}{2x^2-2x+1}\,dx\\
&=\int_0^1 \frac{1}{(2x-1)^2+1}\,dx\\
\end{align}$
Perform the change of variable $y=2x-1$,
$\begin{align}
J&=\frac{1}{2}\int_{-1}^1 \frac{1}{x^2+1}\,dx\\
&=\frac{1}{2}\Big[\arctan x\Big]_{-1}^1\\
&=\frac{1}{2}\times\frac{\pi}{2}\\
&=\boxed{\frac{\pi}{4}}
\end{align}$
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