∫ x^3√(a^2 - x^2) dx
= ∫ x[a^2 - (a^2 - x^2)]√(a^2 - x^2) dx
= ∫ x[a^2√(a^2 - x^2) - (a^2 - x^2)^(3/2)] dx
= (- 1/2)∫ [a^2√(a^2 - x^2) - (a^2 - x^2)^(3/2)] d(a^2 - x^2)
= (- 1/2) * [a^2 * (2/3)(a² - x^2)^(3/2) - (2/5)(a^2 - x^2)^(5/2)] + C
= (- 1/15)(2a^2 + 3x^2)(a^2 - x^2)^(3/2) + C