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.

If you want to continue your solution, then with substitution $t^2=u$
$$I=\int_0^1\dfrac{t}{2t^4-2t^2+1}dt=\dfrac12\int_0^1\dfrac{1}{2u^2-2u+1}du=\int_0^1\dfrac{1}{(2u-1)^2+1}du$$
and then with substitution $2u-1=w$
$$I=\dfrac12\int_{-1}^1\dfrac{1}{w^2+1}dw=\color{blue}{\dfrac{\pi}{4}}$$
