分部積分法
∫xarcsinxdx
=∫arcsinxd(x²/2)
=(x²/2)arcsinx-∫(x²/2)darcsinx
=(x²/2)arcsinx-∫(x²/2)/√(1-x²)dx
=(x²/2)arcsinx+(1/2)∫(-x²)/√(1-x²)dx
=(x²/2)arcsinx+(1/2)∫[(1-x²)-1]/√(1-x²)dx
=(x²/2)arcsinx-(1/2)arcsinx+(1/2)∫√(1-x²)dx ①
又
∫√(1-x²)dx
=x√(1-x²)-∫xd√(1-x²)
=x√(1-x²)-∫[(-x²)/√(1-x²)]dx
=x√(1-x²)-∫[(1-x²+1)/√(1-x²)]dx
=x√(1-x²)-arcsinx-∫√(1-x²)dx
移項(xiàng)后兩邊同除以2得
∫√(1-x²)dx=(1/2)[x√(1-x²)-arcsinx]+2C
代入①得
∫xarcsinxdx
=(x²/2)arcsinx-(1/2)arcsinx+(1/4)[x√(1-x²)-arcsinx]+C
=(1/4)[(2x²-3)arcsinx+x√(1-x²)]+C