计算定积分∫上限负2√ 2下限负2,√ x^2-4/x^3dx

题目

计算定积分∫上限负2√ 2下限负2,√ x^2-4/x^3dx

答案

令x = 2secθ,dx = 2secθtanθ dθ
当x = -2,-2 = 2secθ => secθ = -1 => θ = π
当x = -2√2,-2√2 = 2secθ => secθ = -√2 => θ = 3π/4
∫(-2~-2√2) √(x² - 4)/x³ dx
= ∫(π~3π/4) √(4sec²θ - 4)/(8sec³θ) * (2secθtanθ) dθ
= ∫(π~3π/4) √(4tan²θ) * 1/4 * tanθ/sec²θ dθ
= ∫(π~3π/4) -2tan²θ/sec²θ * 1/4 dθ,消除是根号后有符号∵x