求解∫xarcsin(x/2)dx用分部积分法，谢谢啦

Question

求解∫xarcsin(x/2)dx
用分部积分法，谢谢啦

Answer 1

∫ xarcsin(x/2) dx
= ∫ arcsin(x/2) d(x²/2)
= (1/2)x²arcsin(x/2) - (1/2)∫ x² * 1/√[1 - (x/2)²] * (1/2) dx
= (1/2)x²arcsin(x/2) - (1/2)∫ x²/√(4 - x²) dx
x = 2sinθ,dx = 2cosθ dθ
= (1/2)x²arcsin(x/2) - (1/2)∫ 4sin²θ/(2cosθ) * (2cosθ) dθ
= (1/2)x²arcsin(x/2) - 2∫ (1 - cos2θ)/2 dθ
= (1/2)x²arcsin(x/2) - ∫ dθ + ∫ cos2θ dθ
= (1/2)x²arcsin(x/2) - θ + sinθcosθ + C
= (1/2)x²arcsin(x/2) - arcsin(x/2) + (x/2)√[1 - (x/2)²] + C
= (1/2)x²arcsin(x/2) - arcsin(x/2) + (x/4)√(4 - x²) + C