求定积分∫ (arcsinx)^2dx.上限1,下限0

令arcsinx = t。
∫(arcsinx)²dx {0→1}
= ∫t²d(sint) {0→π/2}
= t²sint {0→π/2} - 2∫tsintdt {0→π/2}
= π²/4 + 2∫td(cost) {0→π/2}
= π²/4 + 2tcost {0→π/2} - 2∫costdt {0→π/2}
= π²/4 - 2sint {0→π/2}
= π²/4 - 2.

t=arcsinx
x=sint
∫ t^2dsintdt
=-∫ t^2costdt
上限是 sint=1 t=pai/2
下限是sint=0 t=0
=-(0-0)=0

∫ (arcsinx)^2dx
令arcsinx=t,dx=costdt
∫ t^2costdt