∫√x(x-3)dx求解答过程和答案阿.

Question

∫√x(x-3)dx
求解答过程和答案阿.

Answer 1

∫ √[x(x - 3)] dx
= ∫ √(x² - 3x) dx
= ∫ √[(x - 3/2)² - 9/4] dx
令x - 3/2 = (3/2)secθ,dx = (3/2)secθtanθ dθ
secθ = (2x - 3)/3,tanθ = √(sec²θ - 1) = (2/3)√(x² - 3x)
= ∫ √[(9/4)sec²θ - 9/4] • (3/2)secθtanθ dθ
= (9/4)∫ tan²θsecθ dθ
= (9/4)∫ (sec²θ - 1)secθ dθ
= (9/4)∫ (sec³θ - secθ) dθ
= (9/4)(1/2)[secθtanθ + ln|secθ + tanθ|] - (9/4)ln|secθ + tanθ| + C
= (9/4)(1/2)(secθtanθ) + (9/4)(-1/2)ln|secθ + tanθ| + C
= (9/8)(1/3)(2x - 3)(2/3)√(x² - 3x) - (9/8)ln|(2x - 3)/3 + (2/3)√(x² - 3x)| + C
= (1/4)(2x - 3)√(x² - 3x) - (9/8)ln|2x - 3 + 2√(x² - 3x)| + C
______________________________________________________________
另一个方法就是令x = 3sec²θ了