Evaluate: ∫ sin 2x/(sin^4x+cos^4x)dx, x ∈[0, π/2]

Question

Evaluate:  

\(\int\limits_0^{\pi/2}\cfrac{sin\,2x}{(sin^4x+cos^4x)}dx\)

∫ sin 2x/(sin⁴x+cos⁴x)dx, x ∈[0, π/2]

Answer 1

Let \(I=\int\limits_0^{\pi/2}\cfrac{sin\,2x}{(sin^4x+cos^4x)}dx\)

Using sin 2x =2 sin x cos x, we get