∫xarcsinxdx用分部积分法
=1/2*∫arcsinxdx^2
=1/2*x^2*arcsinx-1/2∫x^2darcsinx
=1/2*x^2*arcsinx-1/2∫x^2/√(1-x^2)dx
令x=sint，那么，
∫x^2/√(1-x^2)dx
=∫(sint)^2/costdsint
=∫(sint)^2dt
=∫(1-cos2t)/2dt
=1/2t-1/4sin2t+C=1/2t-1/2sint*cost+C
又x=sint，则t=arcsinx，cost=√(1-x^2)，那么
∫x^2/√(1-x^2)dx=1/2t-1/2sint*cost+C=1/2arcsinx-1/2*x*√(1-x^2)+C
那么∫xarcsinxdx=1/2*x^2*arcsinx-1/2∫x^2/√(1-x^2)dx
=1/2*x^2*arcsinx-1/4arcsinx+1/4*x*√(1-x^2)+C