∫(0->π/2) sin^3t dt
=-∫(0->π/2)sin^2tdcost
=-∫(0->π/2)(1-cos^2t)dcost
=-∫(0->π/2)dcost+∫(0->π/2)cos^2t dcost
=-cost|(0->π/2)+cos^3t/3 |(0->π/2)
=-(0-1)+(0-1/3)
=1-1/3
=2/3