求教定积分问题:积分限是0到2,求I=∫x(√2x-x^2)dx是陈文灯的视频里面的一个练习题,他没写答案.自己弄不出来啊积分限是0到2,求I=∫x(√2x-x^2)dx也许我打的有点问题,是I=∫(0,2)x√(2x-x^2)dx

Question

求教定积分问题:积分限是0到2,求I=∫x(√2x-x^2)dx
是陈文灯的视频里面的一个练习题,他没写答案.自己弄不出来啊
积分限是0到2,求I=∫x(√2x-x^2)dx
也许我打的有点问题,是I=∫(0,2)x√(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