править
Пусть
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
[
a
,
b
]
{\displaystyle [a,b]}
n
{\displaystyle n}
x
1
,
…
,
x
n
−
1
{\displaystyle x_{1},\dots ,x_{n-1}}
a
=
x
0
<
x
1
<
⋯
<
x
n
−
1
<
x
n
=
b
{\displaystyle a=x_{0}<x_{1}<\dots <x_{n-1}<x_{n}=b}
τ
=
{
x
i
}
i
=
0
n
{\displaystyle \tau =\{x_{i}\}_{i=0}^{n}}
[
a
,
b
]
{\displaystyle [a,b]}
Δ
x
i
=
x
i
−
x
i
−
1
{\displaystyle \Delta x_{i}=x_{i}-x_{i-1}}
[
x
i
−
1
,
x
i
]
{\displaystyle [x_{i-1},x_{i}]}
λ
=
max
i
=
1
,
n
¯
Δ
x
i
{\displaystyle \lambda =\max {i={\overline {1,n}}}\Delta x_{i}}
τ
{\displaystyle \tau }
На каждом частичном отрезке
[
x
i
−
1
,
x
i
]
{\displaystyle [x_{i-1},x_{i}]}
ξ
i
∈
[
x
i
−
1
,
x
i
]
{\displaystyle \xi _{i}\in [x_{i-1},x_{i}]}
Составим интегральную сумму для функции
f
{\displaystyle f}
τ
:
σ
τ
(
f
)
=
∑
f
(
ξ
i
)
Δ
x
i
{\displaystyle \tau :\sigma _{\tau }(f)=\sum f(\xi _{i})\Delta x_{i}}
Определение. Если
∃
lim
λ
→
0
σ
τ
(
f
)
{\displaystyle \exists \lim _{\lambda \to 0}\sigma _{\tau }(f)}
τ
{\displaystyle \tau }
ξ
i
{\displaystyle \xi _{i}}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)dx}
править
Пусть
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
∀
{\displaystyle \forall }
τ
=
{
x
i
}
i
=
0
n
{\displaystyle \tau =\{x_{i}\}_{i=0}^{n}}
[
a
,
b
]
{\displaystyle [a,b]}
M
i
=
sup
[
x
i
−
1
,
x
i
]
f
(
x
)
{\displaystyle M_{i}=\sup _{[x_{i-1},x_{i}]}f(x)}
m
i
=
inf
[
x
i
−
1
,
x
i
]
f
(
x
)
,
i
=
1
,
n
¯
{\displaystyle m_{i}=\inf _{[x_{i-1},x_{i}]}f(x),i={\overline {1,n}}}
s
τ
=
∑
i
=
1
n
m
i
Δ
x
i
{\displaystyle s_{\tau }=\sum _{i=1}^{n}m_{i}\Delta x_{i}}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
S
τ
=
∑
i
=
1
n
M
i
Δ
x
i
{\displaystyle S_{\tau }=\sum _{i=1}^{n}M_{i}\Delta x_{i}}
Теорема. Пусть функция
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
⇔
∃
lim
λ
→
0
(
S
τ
−
s
τ
)
=
0
{\displaystyle [a,b]\Leftrightarrow \exists \lim _{\lambda \to 0}(S_{\tau }-s_{\tau })=0}
Теорема. Если функция
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
Доказательство.
f
{\displaystyle f}
[
a
,
b
]
⇒
f
{\displaystyle [a,b]\Rightarrow f}
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
Фиксируем
∀
ε
>
0
{\displaystyle \forall \varepsilon >0}
для
ε
b
−
a
>
0
∃
δ
>
0
∀
ξ
,
η
∈
[
a
,
b
]
|
ξ
−
η
|
<
δ
,
|
f
(
ξ
)
−
f
(
η
)
|
<
ε
b
−
a
{\displaystyle {\frac {\varepsilon }{b-a}}>0\quad \exists \delta >0\quad \forall \xi ,\eta \in [a,b]\quad |\xi -\eta |<\delta ,|f(\xi )-f(\eta )|<{\frac {\varepsilon }{b-a}}}
Рассмотрим
∀
τ
=
{
x
i
}
i
=
0
n
{\displaystyle \forall \tau =\{x_{i}\}_{i=0}^{n}}
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
[
a
,
b
]
⇒
f
{\displaystyle [a,b]\Rightarrow f}
∀
[
x
i
−
1
,
x
i
]
i
=
1
,
n
¯
⇒
∃
ξ
i
,
η
i
∈
[
x
i
−
1
,
x
i
]
:
f
(
ξ
i
)
=
inf
[
x
i
−
1
,
x
i
]
f
(
x
)
=
m
i
{\displaystyle \forall [x_{i-1},x_{i}]i={\overline {1,n}}\Rightarrow \exists \xi _{i},\eta _{i}\in [x_{i-1},x_{i}]:f(\xi _{i})=\inf _{[x_{i-1},x_{i}]}f(x)=m_{i}}
f
(
η
i
)
=
sup
[
x
i
−
1
,
x
i
]
f
(
x
)
=
M
i
{\displaystyle f(\eta _{i})=\sup _{[x_{i-1},x_{i}]}f(x)=M_{i}}
∀
i
=
1
,
n
¯
{\displaystyle \forall i={\overline {1,n}}}
Рассмотрим
S
τ
−
s
τ
=
∑
(
M
i
−
m
i
)
Δ
x
i
=
∑
(
f
(
η
i
)
−
f
(
ξ
i
)
)
Δ
x
i
{\displaystyle S_{\tau }-s_{\tau }=\sum (M_{i}-m_{i})\Delta x_{i}=\sum (f(\eta _{i})-f(\xi _{i}))\Delta x_{i}}
Пусть
λ
<
δ
⇒
Δ
x
i
<
δ
∀
i
=
1
,
n
¯
⇒
|
ξ
i
−
η
i
|
<
δ
,
∀
i
=
1
,
n
¯
⇒
|
f
(
ξ
i
)
−
f
(
η
i
)
|
<
ε
b
−
a
,
∀
i
=
1
,
n
¯
{\displaystyle \lambda <\delta \Rightarrow \Delta x_{i}<\delta \forall i={\overline {1,n}}\Rightarrow |\xi _{i}-\eta _{i}|<\delta ,\forall i={\overline {1,n}}\Rightarrow |f(\xi _{i})-f(\eta _{i})|<{\frac {\varepsilon }{b-a}},\forall i={\overline {1,n}}}
⇒
S
τ
−
s
τ
<
ε
b
−
a
∑
Δ
x
i
=
ε
⇒
0
<
S
τ
−
s
τ
<
ε
{\displaystyle \Rightarrow S_{\tau }-s_{\tau }<{\frac {\varepsilon }{b-a}}\sum \Delta x_{i}=\varepsilon \Rightarrow 0<S_{\tau }-s_{\tau }<\varepsilon }
∀
τ
:
λ
<
δ
⇒
∃
lim
λ
→
0
(
S
τ
−
s
τ
)
=
0
⇒
f
{\displaystyle \forall \tau :\lambda <\delta \Rightarrow \exists \lim _{\lambda \to 0}(S_{\tau }-s_{\tau })=0\Rightarrow f}
[
a
,
b
]
{\displaystyle [a,b]}
править
∫
a
b
d
x
=
b
−
a
{\displaystyle \int _{a}^{b}dx=b-a}
f
(
x
)
,
g
(
x
)
{\displaystyle f(x),g(x)}
[
a
,
b
]
{\displaystyle [a,b]}
[
f
(
x
)
+
g
(
x
)
]
{\displaystyle [f(x)+g(x)]}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
(
f
(
x
)
+
g
(
x
)
)
d
x
=
∫
a
b
f
(
x
)
d
x
+
∫
a
b
g
(
x
)
d
x
{\displaystyle \int _{a}^{b}(f(x)+g(x))dx=\int _{a}^{b}f(x)dx+\int _{a}^{b}g(x)dx}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
(
k
⋅
f
(
x
)
)
{\displaystyle (k\cdot f(x))}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
k
f
(
x
)
d
x
=
k
∫
a
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}kf(x)dx=k\int _{a}^{b}f(x)dx}
f
(
x
)
{\displaystyle f(x)}
[
a
,
c
]
{\displaystyle [a,c]}
[
c
,
b
]
{\displaystyle [c,b]}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
f
(
x
)
d
x
=
∫
a
c
f
(
x
)
d
x
+
∫
c
b
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)dx=\int _{a}^{c}f(x)dx+\int _{c}^{b}f(x)dx}
f
(
x
)
{\displaystyle f(x)}
g
(
x
)
{\displaystyle g(x)}
[
a
,
b
]
{\displaystyle [a,b]}
f
(
x
)
≤
g
(
x
)
,
∀
x
∈
[
a
,
b
]
{\displaystyle f(x)\leq g(x),\forall x\in [a,b]}
∫
a
b
f
(
x
)
d
x
≤
∫
a
b
g
(
x
)
d
x
{\displaystyle \int _{a}^{b}f(x)dx\leq \int _{a}^{b}g(x)dx}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
|
f
(
x
)
|
{\displaystyle |f(x)|}
[
a
,
b
]
{\displaystyle [a,b]}
|
∫
a
b
f
(
x
)
d
x
|
≤
∫
a
b
|
f
(
x
)
|
d
x
{\displaystyle |\int _{a}^{b}f(x)dx|\leq \int _{a}^{b}|f(x)|dx}
править
Пусть
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
∀
[
a
,
x
]
⊂
[
a
,
b
]
{\displaystyle \forall [a,x]\subset [a,b]}
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
{\displaystyle F(x)=\int _{a}^{x}f(t)dt}
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
F
(
x
)
{\displaystyle F(x)}
F
(
x
)
{\displaystyle F(x)}
править
Теорема. Пусть
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
F
(
x
)
{\displaystyle F(x)}
[
a
,
b
]
{\displaystyle [a,b]}
Доказательство. Фиксируем
∀
x
∈
[
a
,
b
]
{\displaystyle \forall x\in [a,b]}
Δ
x
:
(
x
+
Δ
x
)
∈
[
a
,
b
]
⇒
F
(
x
+
Δ
x
)
−
F
(
x
)
=
∫
a
x
+
Δ
x
f
(
t
)
d
t
−
∫
a
x
f
(
t
)
d
t
=
∫
x
x
+
Δ
x
f
(
t
)
d
t
⇒
|
Δ
F
(
x
)
|
=
|
∫
x
x
+
Δ
x
f
(
t
)
d
t
|
{\displaystyle \Delta x:(x+\Delta x)\in [a,b]\Rightarrow F(x+\Delta x)-F(x)=\int _{a}^{x+\Delta x}f(t)dt-\int _{a}^{x}f(t)dt=\int _{x}^{x+\Delta x}f(t)dt\Rightarrow |\Delta F(x)|=|\int _{x}^{x+\Delta x}f(t)dt|}
f
{\displaystyle f}
[
a
,
b
]
⇒
f
{\displaystyle [a,b]\Rightarrow f}
[
a
,
b
]
⇒
∃
M
>
0
:
|
f
(
t
)
|
≤
M
{\displaystyle [a,b]\Rightarrow \exists M>0:|f(t)|\leq M}
∀
t
∈
[
a
,
b
]
⇒
|
f
(
t
)
|
≤
M
{\displaystyle \forall t\in [a,b]\Rightarrow |f(t)|\leq M}
∀
t
∈
[
x
,
x
+
Δ
x
]
⇒
|
∫
x
x
+
Δ
x
f
(
t
)
d
t
|
≤
|
∫
x
x
+
Δ
x
|
f
(
t
)
|
d
t
|
≤
M
Δ
x
{\displaystyle \forall t\in [x,x+\Delta x]\Rightarrow |\int _{x}^{x+\Delta x}f(t)dt|\leq |\int _{x}^{x+\Delta x}|f(t)|dt|\leq M\Delta x}
0
≤
|
Δ
F
(
x
)
|
≤
M
Δ
x
{\displaystyle 0\leq |\Delta F(x)|\leq M\Delta x}
Если
Δ
x
→
0
{\displaystyle \Delta x\to 0}
∃
lim
Δ
x
→
0
(
Δ
F
(
x
)
)
=
0
⇒
F
{\displaystyle \exists \lim _{\Delta x\to 0}(\Delta F(x))=0\Rightarrow F}
x
{\displaystyle x}
Теорема. Пусть
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
x
0
∈
[
a
,
b
]
{\displaystyle x_{0}\in [a,b]}
F
(
x
)
{\displaystyle F(x)}
x
0
{\displaystyle x_{0}}
F
′
(
x
)
|
x
0
=
f
(
x
0
)
{\displaystyle \left.F'(x)\right|_{x_{0}}=f(x_{0})}
Теорема (формула Ньютона-Лейбница). Пусть
f
{\displaystyle f}
[
a
,
b
]
⇒
{\displaystyle [a,b]\Rightarrow }
Φ
(
x
)
{\displaystyle \Phi (x)}
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
∫
a
b
f
(
x
)
d
x
=
Φ
(
b
)
−
Φ
(
a
)
{\displaystyle \int _{a}^{b}f(x)dx=\Phi (b)-\Phi (a)}
Доказательство.
f
{\displaystyle f}
[
a
,
b
]
⇒
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
{\displaystyle [a,b]\Rightarrow F(x)=\int _{a}^{x}f(t)dt}
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
{\displaystyle [a,b]}
Φ
{\displaystyle \Phi }
f
(
x
)
{\displaystyle f(x)}
[
a
,
b
]
⇒
F
(
x
)
−
Φ
(
x
)
=
C
∀
x
∈
[
a
,
b
]
⇒
∫
a
x
f
(
t
)
=
Φ
(
x
)
+
C
,
∀
x
∈
[
a
,
b
]
{\displaystyle [a,b]\Rightarrow F(x)-\Phi (x)=C\quad \forall x\in [a,b]\Rightarrow \int _{a}^{x}f(t)=\Phi (x)+C,\forall x\in [a,b]}
при
x
=
a
{\displaystyle x=a}
Φ
(
a
)
+
C
=
0
⇒
C
=
−
Φ
(
a
)
⇒
∫
a
x
f
(
t
)
d
t
=
Φ
(
x
)
−
Φ
(
a
)
,
∀
x
∈
[
a
,
b
]
{\displaystyle \Phi (a)+C=0\Rightarrow C=-\Phi (a)\Rightarrow \int _{a}^{x}f(t)dt=\Phi (x)-\Phi (a),\forall x\in [a,b]}
при
x
=
b
{\displaystyle x=b}
∫
a
b
f
(
t
)
d
t
=
Φ
(
b
)
−
Φ
(
a
)
{\displaystyle \int _{a}^{b}f(t)dt=\Phi (b)-\Phi (a)}
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle [x_{i-1},x_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09cb12a889d47020c8ce7046a2eb60785e00c0b6)
![{\displaystyle \xi _{i}\in [x_{i-1},x_{i}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2479e8663984e1276eee950e8ba659d47ebb73e)
![{\displaystyle M_{i}=\sup _{[x_{i-1},x_{i}]}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/030549b3e521849655f0f6f39176f55a5c256318)
![{\displaystyle m_{i}=\inf _{[x_{i-1},x_{i}]}f(x),i={\overline {1,n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1956096c5b5e2ec46a3f164da1f0bc736a490e05)
![{\displaystyle [a,b]\Leftrightarrow \exists \lim _{\lambda \to 0}(S_{\tau }-s_{\tau })=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e19ba94b7b55bd590a4d3d879725f5b7da24db)
![{\displaystyle [a,b]\Rightarrow \exists \lim _{\lambda \to 0}\sigma _{\tau }(f)=I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eedb033365c8ebec3f1ae77cbb6bbe4a9260e6c2)
![{\displaystyle [a,b]\Rightarrow s_{\tau }\leq {\underline {I}}\leq {\overline {I}}\leq S_{\tau }=0\forall \tau \Rightarrow {\begin{matrix}0\leftarrow [rd]&\leq {\overline {I}}-{\underline {I}}\leq &S_{\tau }-s_{\tau }\leftarrow [ld]\\&0&\end{matrix}}\quad \forall \tau \Rightarrow \exists \lim({\overline {I}}-{\underline {I}})=0\Rightarrow {\overline {I}}-{\underline {I}}=0\Rightarrow {\overline {I}}={\underline {I}}=I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f60de280a40224d026221279d05b4efd1bd1d929)
![{\displaystyle {\begin{matrix}s_{\tau }\leftarrow [rd]&\leq \sigma _{\tau }(f)\leq &S_{\tau }\leftarrow [ld]\\&I&\end{matrix}}\Rightarrow \exists \lim _{\lambda \to 0}\sigma _{\tau }(f)=I\Rightarrow f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb56a39080412b93c04533d5f174ae9d14c9922)
![{\displaystyle [a,b]\Rightarrow f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fefae13306667ed703d4fcac388c599dcfe3de38)
![{\displaystyle {\frac {\varepsilon }{b-a}}>0\quad \exists \delta >0\quad \forall \xi ,\eta \in [a,b]\quad |\xi -\eta |<\delta ,|f(\xi )-f(\eta )|<{\frac {\varepsilon }{b-a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/293ec4ff8c1ad8f1ea6f40aff20401372baf61c3)
![{\displaystyle \forall [x_{i-1},x_{i}]i={\overline {1,n}}\Rightarrow \exists \xi _{i},\eta _{i}\in [x_{i-1},x_{i}]:f(\xi _{i})=\inf _{[x_{i-1},x_{i}]}f(x)=m_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c1d5191b8cb221644a47e5e10428a1754eebe4)
![{\displaystyle f(\eta _{i})=\sup _{[x_{i-1},x_{i}]}f(x)=M_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deec15fdaa0f329a853ea100806ac9ecc8013e0f)
![{\displaystyle [f(x)+g(x)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f598a75db4dcfd38f1b880ba87cf91a648cc1c)
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![{\displaystyle \Delta x:(x+\Delta x)\in [a,b]\Rightarrow F(x+\Delta x)-F(x)=\int _{a}^{x+\Delta x}f(t)dt-\int _{a}^{x}f(t)dt=\int _{x}^{x+\Delta x}f(t)dt\Rightarrow |\Delta F(x)|=|\int _{x}^{x+\Delta x}f(t)dt|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f32dca5b6f1335b8d21cd5d21f6e3421092535)
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![{\displaystyle t\in [x_{0},x_{0}+\Delta x]\Rightarrow |t-x_{0}|\leq |\Delta x|\leq \delta \Rightarrow |f(t)-f(x_{0})|<\varepsilon \Rightarrow |I|={\frac {1}{|\Delta x|}}|\int _{x_{0}}^{x_{0}+\Delta x}(f(t)-f(x_{0}))dt|\leq {\frac {1}{\Delta x}}|\int _{x_{0}}^{x_{0}+\Delta x}|f(t)-f(x_{0})|dt|<{\frac {\varepsilon }{\Delta x}}|\int _{x_{0}}^{x_{0}+\Delta x}dx|=\varepsilon \Rightarrow \lim _{\Delta x\to 0}I=0\Rightarrow \exists \lim {\frac {\Delta F(x_{0})}{\Delta x}}=f(x_{0})\Rightarrow \exists \left.F'(x)\right|_{x_{0}}=f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1cfc6dfd21f785d86600ca954e6d6997d59aed)
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