∫(上限2,下限0)x根号下(2x-x^2)dx求大神解答.弄不懂了.希望详细一点谢谢了担心弄混淆再加一下括号∫(上限2,下限0)x[根号下(2x-x^2）]dx

Question

∫(上限2,下限0)x根号下(2x-x^2)dx求大神解答. 弄不懂了.希望详细一点谢谢了
担心弄混淆 再加一下括号 ∫(上限2,下限0)x[根号下(2x-x^2）]dx

Answer 1

∫(0,2) x√(2x - x²) dx
= ∫(0,2) x√[- (x² - 2x + 1) + 1] dx
= ∫(0,2) x√[1 - (x - 1)²] dx
令x - 1 = sinθ,dx = cosθ dθ
x = 0 --> θ = - π/2
x = 2 --> θ = π/2
= ∫(- π/2,π/2) (1 + sinθ)|cosθ| * cosθ dθ
= ∫(- π/2,π/2) (1 + sinθ)cos²θ dθ
= ∫(- π/2,π/2) cos²θ dθ + ∫(- π/2,π/2) sinθcos²θ dθ
= 2∫(0,π/2) (1 + cos2θ)/2 dθ + ∫(- π/2,π/2) cos²θ d(- cosθ)
= [θ + (1/2)sin2θ] |(0,π/2) - (1/3)[cos³θ] |(- π/2,π/2)
= π/2