求积分∫√(1-x^2)^3dx上限1下限0 分母全在根号内

题目

求积分∫√(1-x^2)^3dx上限1下限0 分母全在根号内

答案

令x = sinθ,dx = cosθ dθx∈[0,1] → θ∈[0,π/2]∫(0~1) (1 - x²)^(3/2) dx= ∫(0~π/2) cos³θ • cosθ dθ= ∫(0~π/2) (cos²θ)² dθ= ∫(0~π/2) [(1/2)(1 + cos2θ)]² dθ...