令z = √x,x = z²
∫ arcsin√x dx
= ∫ arcsinz d(z²)
= z²arcsinz - ∫ z²/√(1 - z²) dz
令z = sinθ,dz = cosθ dθ
= z²arcsinz - ∫ sin²θ/|cosθ| * (cosθ dθ)
= z²arcsinz - ∫ (1 - cos2θ)/2 dθ
= z²arcsinz - (1/2)(θ - sinθcosθ) + C
= z²arcsinz - θ/2 + (1/2)sinθcosθ + C
= z²arcsinz - (1/2)arcsinz + (1/2)z√(1 - z²) + C
= xarcsin√x - (1/2)arcsin√x + (1/2)√x√(1 - x) + C
= (x - 1/2)arcsin√x + (1/2)√(x - x²) + C
