?[0, pi/2](sin(x) - cos^2(x)sin(x)dx)
= ?[0, pi/2](sin(x)dx) - ?[0, pi/2](cos^2(x)sin(x)dx)
Looking only at the second integral
J = ?[0, pi/2](cos^2(x)sin(x)dx
substitution:
u = cosx
du = -sin(x)dx
J = ?[0, pi/2](-u^2)du
J = (-u^3)/3|[0, pi/2]
J = (-cos^3(x))/3|[0, pi/2]