>>463
I = ∫[x=0,π/2] log(sin(x)) dx = ∫[x=0,π/2] log(cos(x)) dx

2I = ∫[x=0,π/2] log(sin(x)) dx + ∫[x=0,π/2] log(cos(x)) dx
= ∫[x=0,π/2] log((1/2)sin(2x)) dx
= ∫[x=0,π/2] log(sin(2x)) dx - (π/2)log2

J = ∫[x=0,π/2] log(sin(2x)) dx
= (1/2)∫[t=0,π] log(sin(t)) dt (t=2x)
= (1/2)∫[x=0,π/2] log(sin(x)) dx + (1/2)∫[x=π/2,π] log(sin(x)) dx
= (1/2)∫[x=0,π/2] log(sin(x)) dx + (1/2)∫[x=0,π/2] log(cos(s)) ds (s=x-π/2)
= I

t = -(π/2)log2