∫dx／根号4-9x^2

题目

∫dx／根号4-9x^2

答案

要使√(4 - 9x²) = √[4 - 4(9/4*x²)] = √[4 - 4(3x/2)²] = √(4 - 4sin²θ) = √(4cos²θ) = 2cosθ
令3x/2 = sinθ,dx = (2/3)cosθdθ
∫ dx/√(4 - 9x²)
= ∫ (2/3)cosθ/(2cosθ) dθ
= (1/3)∫ dθ = (1/3)θ + C
= (1/3)arcsin(3x/2) + C