Integral of arcsin(x)/sqrt(1-x^2) (substitution)

Integral of arcsin(x)/sqrt(1-x^2) (substitution) ∫ arcsin(x) / √(1 - x²) dx Step 1: Substitution Let u = arcsin(x). Then: x = sin(u), and dx = cos(u) du. Since x = sin(u), we also have: 1 - x² = 1 - sin²(u) = cos²(u), so √(1 - x²) = cos(u). Rewrite the integral: ∫ arcsin(x) / √(1 - x²) dx = ∫ u / cos(u) * cos(u) du = ∫ u du. Step 2: Integrate ∫ u du = (u² / 2). Step 3: Back-substitute u = arcsin(x) Substitute back u = arcsin(x): (arcsin(x))² / 2 + C. Final Answer: ∫ arcsin(x) / √(1 - x²) dx = (arcsin(x))² / 2 + C.