牛顿-莱布尼茨公式

前置知识:黎曼积分的概念

牛顿-莱布尼茨公式

f f f [ a , b ] [a,b] [a,b]上可积,令

F ( x ) = ∫ a x f ( t ) d t F(x)=\int_a^xf(t)dt F(x)=axf(t)dt

(1) F F F [ a , b ] [a,b] [a,b]上连续
(2)若 f f f在点 x 0 ∈ [ a , b ] x_0\in[a,b] x0[a,b]处连续,则 F F F x 0 x_0 x0处可导,且 F ′ ( x 0 ) = f ( x 0 ) F'(x_0)=f(x_0) F(x0)=f(x0)
(3)若 f f f [ a , b ] [a,b] [a,b]上连续,则 F F F f f f [ a , b ] [a,b] [a,b]上的一个原函数。如果 G G G f f f的任意一个原函数,则有
∫ a b f ( x ) d x = G ( b ) − G ( a ) \int_a^bf(x)dx=G(b)-G(a) abf(x)dx=G(b)G(a)

证明:
(1)因为 f f f [ a , b ] [a,b] [a,b]上可积,所以 f f f [ a , b ] [a,b] [a,b]上有界。令 M M M ∣ f ( x ) ∣ |f(x)| f(x)的最大值,任取 x 0 ∈ [ a , b ] x_0\in[a,b] x0[a,b],当 x ∈ [ a , b ] x\in[a,b] x[a,b]时,有

∣ F ( x ) − F ( x 0 ) ∣ = ∣ ∫ a x f ( t ) d t − ∫ a x 0 f ( t ) d t ∣ |F(x)-F(x_0)|=|\int_a^xf(t)dt-\int_a^{x_0}f(t)dt| F(x)F(x0)=axf(t)dtax0f(t)dt

= ∣ ∫ x 0 x f ( t ) d t ∣ ≤ M ∣ ∫ x 0 x d x ∣ = M ∣ x − x 0 ∣ =|\int_{x_0}^xf(t)dt|\leq M|\int_{x_0}^xdx|=M|x-x_0| =x0xf(t)dtMx0xdx=Mxx0

\qquad 由连续函数的定义,当 x → x 0 x\to x_0 xx0时, ∣ x − x 0 ∣ → 0 |x-x_0|\to 0 xx00 M ∣ x − x 0 ∣ → 0 M|x-x_0|\to 0 Mxx00,所以 F F F在点 x 0 x_0 x0处连续

\qquad 因为 x 0 x_0 x0可以取 [ a , b ] [a,b] [a,b]上的任何值,所以 F F F [ a , b ] [a,b] [a,b]上连续

(2)依题意, x 0 x_0 x0 f f f的连续点,则 ∀ ε > 0 , ∃ δ > 0 \forall\varepsilon>0,\exist\delta>0 ε>0,δ>0,当 t ∈ [ a , b ] t\in[a,b] t[a,b] ∣ t − x 0 ∣ < δ |t-x_0|<\delta tx0<δ时,都有

∣ f ( t ) − f ( x 0 ) ∣ < ε |f(t)-f(x_0)|<\varepsilon f(t)f(x0)<ε

\qquad 于是,当 x ∈ [ a , b ] x\in[a,b] x[a,b] ∣ x − x 0 ∣ < δ |x-x_0|<\delta xx0<δ时,

∣ F ( x ) − F ( x 0 ) x − x 0 − f ( x 0 ) ∣ = ∣ 1 x − x 0 ∫ x 0 x [ f ( t ) − f ( x 0 ) ] d t ∣ < ∣ 1 x − x 0 ∫ x 0 x ε d t ∣ = ε |\dfrac{F(x)-F(x_0)}{x-x_0}-f(x_0)|=|\dfrac{1}{x-x_0}\int_{x_0}^x[f(t)-f(x_0)]dt|<|\dfrac{1}{x-x_0}\int_{x_0}^x\varepsilon dt|=\varepsilon xx0F(x)F(x0)f(x0)=xx01x0x[f(t)f(x0)]dt<xx01x0xεdt=ε

\qquad 由此可得

F ′ ( x 0 ) = F ( x ) − F ( x 0 ) x − x 0 = f ( x 0 ) F'(x_0)=\dfrac{F(x)-F(x_0)}{x-x_0}=f(x_0) F(x0)=xx0F(x)F(x0)=f(x0)

(3)因为 f f f [ a , b ] [a,b] [a,b]上连续,由 ( 2 ) (2) (2) F ( x ) F(x) F(x) f f f [ a , b ] [a,b] [a,b]上的一个原函数。

\qquad G G G f f fD 的任意一个原函数,则

[ G ( x ) − F ( x ) ] ′ = G ′ ( x ) − F ′ ( x ) = f ( x ) − f ( x ) = 0 [G(x)-F(x)]'=G'(x)-F'(x)=f(x)-f(x)=0 [G(x)F(x)]=G(x)F(x)=f(x)f(x)=0

\qquad 所以 G ( x ) − F ( x ) = C G(x)-F(x)=C G(x)F(x)=C,由此可得 ∀ x ∈ [ a , b ] \forall x\in[a,b] x[a,b],有

∫ a x f ( t ) d t = F ( x ) = F ( x ) − F ( a ) = G ( x ) − G ( a ) \int_a^xf(t)dt=F(x)=F(x)-F(a)=G(x)-G(a) axf(t)dt=F(x)=F(x)F(a)=G(x)G(a)

\qquad 特别地,有

∫ a b f ( t ) d t = G ( b ) − G ( a ) \int_a^bf(t)dt=G(b)-G(a) abf(t)dt=G(b)G(a)

\qquad 这个式子就是牛顿-莱布尼茨公式,这是一种用被积函数的原函数来求定积分的方法。


例题

f f f [ a , b ] [a,b] [a,b]上连续, u ( x ) u(x) u(x) v ( x ) v(x) v(x) [ a , b ] [a,b] [a,b]上可导,且 u ( x ) u(x) u(x) v ( x ) v(x) v(x)的值域包含于 [ a , b ] [a,b] [a,b],求下列函数的导数:

G ( x ) = ∫ v ( x ) u ( x ) f ( t ) d t G(x)=\int_{v(x)}^{u(x)}f(t)dt G(x)=v(x)u(x)f(t)dt

解:
\qquad F ( x ) = ∫ a x f ( t ) d t F(x)=\int_a^xf(t)dt F(x)=axf(t)dt,则 F ′ ( u ) = f ( u ) F'(u)=f(u) F(u)=f(u),所以

G ( x ) = ∫ a u ( x ) f ( t ) d t − ∫ a v ( x ) f ( t ) d t = F ( u ( x ) ) − F ( v ( x ) ) G(x)=\int_a^{u(x)}f(t)dt-\int_a^{v(x)}f(t)dt=F(u(x))-F(v(x)) G(x)=au(x)f(t)dtav(x)f(t)dt=F(u(x))F(v(x))

\qquad 那么

G ′ ( x ) = F ′ ( u ( x ) ) u ′ ( x ) − F ′ ( v ( x ) ) v ′ ( x ) = f ( u ( x ) ) u ′ ( x ) − f ( v ( x ) ) v ′ ( x ) G'(x)=F'(u(x))u'(x)-F'(v(x))v'(x)=f(u(x))u'(x)-f(v(x))v'(x) G(x)=F(u(x))u(x)F(v(x))v(x)=f(u(x))u(x)f(v(x))v(x)

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