求不定积分∫x²/（a²-x²）^½dx

求不定积分∫x²/（a²-x²）^½dx
求不定积分∫x²/（a²-x²）^½dx

求不定积分∫x²/（a²-x²）^½dx
令x = a · sinθ,dx = acosθdθ
∫ x²/√(a² - x²) dx = ∫ a²sin²θ/√(a² - a²sin²θ) · acosθ dθ
= ∫ a²sin²θ/(acosθ) · acosθ dθ
= a² · ∫ (1 - cos2θ)/2 dθ
= a²/2 · (θ - 1/2 · sin2θ) + C
= (a²/2)θ - (a²/2)(sinθcosθ) + C
= (a²/2)arcsin(x/a) - (a²/2) · x/a · √(a² - x²)/a + C
= (a²/2)arcsin(x/a) - (x/2)√(a² - x²) + C