Integral of (sin^2x)/root(1-cosx)

∫(sin²(x))/√(1 - cos(x)) dx

∫(1 - cos²(x))/√(1 - cos(x)) dx

∫(1 + cos(x))√(1 - cos(x)) dx

Recall that sin²(x/2) = (1 - cos(x))/2 and that cos²(x/2) = (1 + cos(x))/2

∫(2*cos²(x/2))*(√2*sin(x/2)) dx

2√2*∫cos²(x/2)*sin(x/2) dx

u = cos(x/2)

-2 du = sin(x/2) dx

-4√2*∫u² du = (-4√2)/3 * u³ + C

= (-4√2)/3 * cos³(x/2) + C