Let I = ∫(2 log sin x - log sin 2x) dx, for x ∈ [0, π/2]
= ∫{2 log sin x - log(2sin x.cos x)} dx, for x ∈ [0, π/2]
= ∫{2 log sin x - log2 - log sin x - log cos x} dx, for x ∈ [0, π/2]
= ∫(log sin x - log2 - log cos x) dx, for x ∈ [0, π/2]
= ∫(log sin x dx - ∫log2 dx - ∫log cos x dx), for x ∈ [0, π/2]
= ∫{(log sin x dx - log2∫dx - ∫log cos((π/2) - x)dx}, for x ∈ [0, π/2]