第2个回答 2020-03-23
∫(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