题目

求上限是2,下限是1的(根号x的平方-1/x)的定积分

提问时间：2020-12-27

答案

令x = secθ,dx = secθtanθ dθ
√(x² - 1) = √(sec²θ - 1) = |tanθ| = tanθ,∵x ≥ 1
当x = 1,θ = 0；当x = 2,θ = π/3
∫(1,2) √(x² - 1)/x dx
= ∫(0,π/3) tanθ/secθ* secθtanθ dθ
= ∫(0,π/3) tan²θ dθ
= ∫(0,π/3) (sec²θ - 1) dθ
= [tanθ - θ] |(0,π/3)
= √3 - π/3