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

Question

Evaluate:

\(\int_0^{4\pi} \cfrac{sin\,2x}{sin^4x+cos^4x}.dx\)

∫ (sin 2x)/(sin⁴ x + cos⁴ x).dx, x ∈ [0,4π]

Answer 1

Let \(I\) = \(\int_0^{4\pi} \cfrac{sin\,2x}{sin^4x+cos^4x}.dx\)

Dividing each term by cos⁴x, we get

Put tan²x = t   \(\therefore\) 2 tan x sec²x dx = dt