F(t)=∫上标t下标1dy∫上标t下标y f(x)dx,且f(x)为连续函数则F'(2)=?

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F(t)=∫上标t下标1dy∫上标t下标y f(x)dx,且f(x)为连续函数则F'(2)=?

F(t)=∫上标t下标1dy∫上标t下标y f(x)dx,且f(x)为连续函数则F'(2)=?
F(t)=∫上标t下标1dy∫上标t下标y f(x)dx,且f(x)为连续函数则F'(2)=?

F(t)=∫上标t下标1dy∫上标t下标y f(x)dx,且f(x)为连续函数则F'(2)=?
ƒ(x)不知道嘛,所以先交换次序
1 ≤ y ≤ t,y ≤ x ≤ t 1 ≤ x ≤ t,1 ≤ y ≤ x
F(t) = ∫(1→t) dy ∫(y→t) ƒ(x) dx
==> F(t) = ∫(1→t) ƒ(x) dx ∫(1→x) dy
==> F(t) = ∫(1→t) ƒ(x) · (x - 1) dx
==> F'(t) = (t - 1) · ƒ(t)
==> F'(2) = (2 - 1) · ƒ(2) = ƒ(2)