Question

定积分0-2 x乘以根号下2x-x^2dx=

Answer 1

∫(0→2) x√(2x - x²) dx= ∫(0→2) x√[1 - (x - 1)²] dx 令x - 1 = sinθ,dx = cosθdθ,第二换元积分法
当x = 0,θ = - π/2
当x = 2,θ = π/2
= ∫(- π/2→π/2) (1 + sinθ) * cos²θ dθ= ∫(- π/2→π/2) cos²θ dθ + ∫(- π/2→π/2) sinθcos²θ dθ, 第一个偶函数,第二个奇函数
= 2∫(0→π/2) cos²θ dθ + 0
= 2 * 1/2 * π/2
= π/2

Answer 2

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∫(0→2)
x√(2x
-
x²)
dx
=
∫(0→2)
x√[1
-
(x
-
1)²]
dx
令x
-
1
=
sinθ,dx
=
cosθdθ,第二换元积分法
当x
=
0,θ
=
-
π/2
当x
=
2,θ
=
π/2
=
∫(-
π/2→π/2)
(1
+
sinθ)
*
cos²θ
dθ
=
∫(-
π/2→π/2)
cos²θ
dθ
+
∫(-
π/2→π/2)
sinθcos²θ
dθ,第一个偶函数,第二个奇函数
=
2∫(0→π/2)
cos²θ
dθ
+
0
=
2
*
1/2
*
π/2
=
π/2