Question

∫x^2√(a^2-x^2)dx

Answer 1

令x = asinθ，dx = acosθ dθ，√(a² - x²) = √(a² - a²sin²θ) = √(a²cos²θ) = acosθ
∫ x²√(a² - x²) dx
= ∫ a²sin²θ • acosθ • acosθ dθ
= a⁴∫ sin²θcos²θ dθ
= a⁴∫ (1/2 • sin2θ)² dθ
= (a⁴/4)(1/2)∫ (1 - cos4θ) dθ
= (a⁴/8)(θ - 1/4 • sin4θ) + C
= (a⁴/8)arcsin(x/a) - (a⁴/8)sinθcosθ(cos²θ - sin²θ) + C
= (a⁴/8)arcsin(x/a) - (a⁴/8)(x/a)[√(a² - x²)/a][(a² - x²)/a² - x²/a²] + C
= (a⁴/8)arcsin(x/a) + (x/8)(2x² - a²)√(a² - x²) + C