定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2

题目

定积分:分母是X乘上根号下(x^2+4),分子是一.上限是2根号3,下限是2

答案

换元法:
x=2tanθ,由于x∈[2,2√3],从而θ∈[π/4,π/3],
√(x²+4)=2/cosθ,dx=2dθ/cos²θ,
从而∫[2,2√3]dx/(x√(x²+4))
=1/2∫[π/4,π/3]dθ/sinθ
=-1/4*ln2+1/2*ln(2+√2)-1/4*ln3
凑微分法:
∫[2,2√3]dx/(x√(x²+4))
=∫[2,2√3]dx/(x²√(1+4/x²))
=∫[2,2√3]x^(-2)dx/√(1+4/x²)
=-1/2∫[2,2√3]d(2/x)/√(1+(2/x)²)
=-1/2ln(2/x+√(1+(2/x)²))[2,2√3]
=-1/4*ln2+1/2*ln(2+√2)-1/4*ln3