x乘以arcsinx的定积分怎么算(积分区间在0-1)?
发布网友
发布时间：2022-12-30 15:12
共2个回答
热心网友
时间：2023-10-30 19:16
=0.5∫(0→1)arcsinxd(x²)
=0.5x²arcsinx∫(0→1)-0.5∫(0→1)x²d(arcsinx)
=π/4-0.5∫(0→1)x²/根号(1-x²)dx
=π/4-0.5∫(0→1)1/根号(1-x²)dx+0.5∫(0→1)根号(1-x²)dx
↑
这一部分是半径为1的四分之一圆面积
=π/4-0.5arcsinx(0→1)+π/8
=π/4-π/4+π/8
=π/8
热心网友
时间：2023-10-30 19:17
∫[0,-1]xarcsinxdx
x=sinu, dx=cosxdx x=0,u=0 x=-1 u=-π/2
=∫[0,-π/2]sinu *ucosu
=(-1/4)∫[0,-π/2]udcos2u
=(-1/4)ucos2u|[0,-π/2]+(1/4)∫[0,-π/2] cos2u
=π/8+(1/8)sin2u[0,-π/2]
=π/8
