[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
{\displaystyle \textstyle f}
x
∈
[
a
,
b
]
{\displaystyle \textstyle x\in \left[a,b\right]}
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
{\displaystyle \textstyle F(x)=\int \limits _{a}^{x}f(t)\,dt}
x
∈
[
a
,
b
]
{\displaystyle \textstyle x\in \left[a,b\right]}
f
{\displaystyle \textstyle f}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
{\displaystyle \textstyle f}
[
a
,
x
]
{\displaystyle \left[a,x\right]}
f
{\displaystyle \textstyle f}
a
⩽
x
⩽
b
{\displaystyle a\leqslant x\leqslant b}
F
(
a
)
=
∫
a
a
f
(
t
)
d
t
=
0
{\displaystyle F(a)=\int \limits _{a}^{a}f(t)\,dt=0}
F
(
b
)
=
∫
a
b
f
(
t
)
d
t
{\displaystyle F(b)=\int \limits _{a}^{b}f(t)\,dt}
F
{\displaystyle \textstyle F}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
x
,
x
+
h
∈
[
a
,
b
]
{\displaystyle x,x+h\in \left[a,b\right]}
F
(
x
+
h
)
−
F
(
x
)
=
∫
a
(
x
+
h
)
f
(
t
)
d
t
−
∫
a
x
f
(
t
)
d
t
=
∫
x
(
x
+
h
)
f
(
t
)
d
t
{\displaystyle F(x+h)-F(x)=\int \limits _{a}^{(x+h)}f(t)\,dt-\int \limits _{a}^{x}f(t)\,dt=\int \limits _{x}^{(x+h)}f(t)\,dt}
K
=
s
u
p
|
f
(
t
)
|
,
a
⩽
t
⩽
b
{\displaystyle K=sup|f(t)|,a\leqslant t\leqslant b}
|
F
(
x
+
h
)
−
F
(
x
)
|
⩽
|
∫
x
(
x
+
h
)
f
(
t
)
d
t
|
⩽
K
|
h
|
→
0
,
h
→
0
{\displaystyle |F(x+h)-F(x)|\leqslant {\bigg |}\int \limits _{x}^{(x+h)}f(t)\,dt{\bigg |}\leqslant K|h|\to 0,h\to 0}
F
{\displaystyle \textstyle F}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
{\displaystyle \textstyle f}
f
{\displaystyle \textstyle f}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
Файл:График.JPG
f
{\displaystyle \textstyle f}
a
A
B
x
{\displaystyle \textstyle aABx}
F
(
x
)
{\displaystyle \textstyle F(x)}
F
(
x
+
h
)
−
F
(
x
)
{\displaystyle \textstyle F(x+h)-F(x)}
x
B
C
(
x
+
h
)
{\displaystyle \textstyle xBC(x+h)}
f
{\displaystyle \textstyle f}
x
{\displaystyle \textstyle x}
f
{\displaystyle \textstyle f}
x
−
d
{\displaystyle \textstyle x-d}
h
→
0
{\displaystyle h\to 0}
Хәзер
f
{\displaystyle \textstyle f}
[
a
,
x
]
{\displaystyle \left[a,x\right]}
x
∈
[
a
,
x
]
{\displaystyle x\in \left[a,x\right]}
F
{\displaystyle \textstyle F}
F
′
(
x
)
=
f
(
x
)
{\displaystyle \textstyle F'(x)=f(x)}
x
{\displaystyle \textstyle x}
F
(
x
+
h
)
−
F
(
x
)
h
=
1
h
∫
x
x
+
h
f
(
t
)
d
t
=
1
h
∫
x
x
+
h
(
f
(
x
)
+
η
(
t
)
)
d
t
{\displaystyle {\dfrac {F(x+h)-F(x)}{h}}={\dfrac {1}{h}}\int \limits _{x}^{x+h}f(t)\,dt={\dfrac {1}{h}}\int \limits _{x}^{x+h}(f(x)+\eta (t))\,dt}
=
1
h
∫
x
x
+
h
f
(
x
)
d
t
+
1
h
∫
x
x
+
h
η
(
t
)
d
t
=
f
(
x
)
+
o
{\displaystyle ={\dfrac {1}{h}}\int \limits _{x}^{x+h}f(x)\,dt+{\dfrac {1}{h}}\int \limits _{x}^{x+h}\eta (t)\,dt=f(x)+o}
h
→
0
{\displaystyle h\to 0}
f
(
t
)
=
f
(
x
)
+
η
(
t
)
{\displaystyle \textstyle f(t)=f(x)+\eta (t)}
f
(
x
)
{\displaystyle \textstyle f(x)}
t
{\displaystyle \textstyle t}
∫
x
x
+
h
f
(
x
)
d
t
=
f
(
x
)
h
{\displaystyle \textstyle \int \limits _{x}^{x+h}f(x)\,dt=f(x)h}
f
{\displaystyle \textstyle f}
x
{\displaystyle \textstyle x}
ε
>
0
{\displaystyle \textstyle \varepsilon >0}
δ
{\displaystyle \textstyle \delta }
|
x
−
t
|
<
δ
{\displaystyle \textstyle |x-t|<\delta }
|
η
(
t
)
|
<
ε
{\displaystyle \textstyle |\eta (t)|<\varepsilon }
|
1
h
∫
x
x
+
h
η
(
t
)
d
t
|
⩽
1
|
h
|
|
h
|
ε
=
ε
,
|
h
|
<
δ
{\displaystyle \left|{\dfrac {1}{h}}\int \limits _{x}^{x+h}\eta (t)\,dt\right|\leqslant {\dfrac {1}{|h|}}|h|\varepsilon =\varepsilon ,|h|<\delta }
h
→
0
{\displaystyle \textstyle h\to 0}
h
→
0
{\displaystyle \textstyle h\to 0}
F
{\displaystyle \textstyle F}
x
{\displaystyle \textstyle x}
F
{\displaystyle \textstyle F}
x
=
a
,
b
{\displaystyle \textstyle x=a,b}
f
{\displaystyle \textstyle f}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
F
(
x
)
=
∫
a
x
f
(
t
)
d
t
{\displaystyle F(x)=\int \limits _{a}^{x}f(t)\,dt}
f
(
x
)
:
F
′
(
x
)
=
f
(
x
)
,
a
⩽
x
⩽
b
{\displaystyle \textstyle f(x):F'(x)=f(x),a\leqslant x\leqslant b}
F
(
x
)
{\displaystyle \textstyle F(x)}
[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
{\displaystyle \textstyle f}
Барроу теоремасы дип атала.
[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
{\displaystyle \textstyle f}
Φ
{\displaystyle \textstyle \Phi }
[
a
,
b
]
{\displaystyle \left[a,b\right]}
f
(
x
)
{\displaystyle \textstyle f(x)}
Φ
(
x
)
=
F
(
x
)
+
C
{\displaystyle \textstyle \Phi (x)=F(x)+C}
C
{\displaystyle \textstyle C}
x
=
a
{\displaystyle \textstyle x=a}
F
(
a
)
=
0
{\displaystyle \textstyle F(a)=0}
Φ
(
a
)
=
C
{\displaystyle \textstyle \Phi (a)=C}
F
(
x
)
=
Φ
(
x
)
−
Φ
(
a
)
{\displaystyle \textstyle F(x)=\Phi (x)-\Phi (a)}
∫
a
b
f
(
x
)
d
x
=
F
(
b
)
{\displaystyle \int \limits _{a}^{b}f(x)\,dx=F(b)}
∫
a
b
f
(
x
)
d
x
=
Φ
(
b
)
−
Φ
(
a
)
=
Φ
(
x
)
|
x
=
a
x
=
b
{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\Phi (b)-\Phi (a)=\Phi (x){\bigg |}{\begin{matrix}x=a\\\\x=b\end{matrix}}}
.gif)
![{\displaystyle \left[a,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f30926fb280a9fdf66fd931e14d4363cb824feaa)
![{\displaystyle \textstyle x\in \left[a,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26167a01437e354daafca04879c926ceb9845d49)
![{\displaystyle \left[a,x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/853571555b7143e7d7218bf35a1e739c05153b2d)
![{\displaystyle x,x+h\in \left[a,b\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd067d069e553f45ed6b4febcda71e76f3724dbf)
![{\displaystyle x\in \left[a,x\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8b14d135fa2b135ebac11566907fa56d2ae89a)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle \forall x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d2f7224adf539aa2be5486d0d90a8b4d3e93cc)
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
