∫x×［（2ax－x×x）½］dx＝?

∫x×［（2ax－x×x）½］dx＝?
∫x×［（2ax－x×x）½］dx＝?

∫x×［（2ax－x×x）½］dx＝?
令x - a = a · sinθ,dx = a · cosθ dθ
∫ x√(2ax - x²) dx
= ∫ x√[a² - (x - a)²] dx
= ∫ a(sinθ + 1) · acosθ · acosθ dθ
= a³∫ cos²θsinθ dθ + a³∫ cos²θ dθ
= - a³∫ cos²θ d(cosθ) + (a³/2)∫ (1 + cos2θ) dθ
= - a³ · (1/3)cos³θ + (a³/2)(θ + 1/2 · sin2θ) + C
= - (a³/3)[√(2ax - x²)/a]³ + (a³/2)[arcsin((x - a)/a) + (x - a)/a · √(2ax - x²)/a] + C
= - (1/6)(3a² + ax - 2x²)√(2ax - x²) + (a³/2)arcsin[(x - a)/a] + C