Let
I
{\displaystyle {}I}
denote a real interval, let
f
:
I
⟶
R
{\displaystyle f\colon I\longrightarrow \mathbb {R} }
denote a
Riemann-integrable function,
and let
a
∈
I
{\displaystyle {}a\in I}
.
Then the function
I
⟶
R
,
x
⟼
∫
a
x
f
(
t
)
d
t
,
{\displaystyle I\longrightarrow \mathbb {R} ,x\longmapsto \int _{a}^{x}f(t)\,dt,}
is called the
integral function for
f
{\displaystyle {}f}
for the starting point
a
{\displaystyle {}a}
.
This function is also called the indefinite integral .
The
x
{\displaystyle {}x}
from the theorem is
x
0
{\displaystyle {}x_{0}}
in the animation, and
x
+
h
{\displaystyle {}x+h}
in the theorem is the moving
x
{\displaystyle {}x}
in the animation. The moving point
z
{\displaystyle {}z}
in the animation is a point which exists by the mean value theorem of definite integrals, applied to
x
0
{\displaystyle {}x_{0}}
and
x
{\displaystyle {}x}
.
The following statement is called Fundamental theorem of calculus .
Let
x
{\displaystyle {}x}
be fixed. The
difference quotient
is
F
(
x
+
h
)
−
F
(
x
)
h
=
1
h
(
∫
a
x
+
h
f
(
t
)
d
t
−
∫
a
x
f
(
t
)
d
t
)
=
1
h
∫
x
x
+
h
f
(
t
)
d
t
.
{\displaystyle {}{\frac {F(x+h)-F(x)}{h}}={\frac {1}{h}}{\left(\int _{a}^{x+h}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right)}={\frac {1}{h}}\int _{x}^{x+h}f(t)\,dt\,.}
We have to show that for
h
→
0
{\displaystyle {}h\rightarrow 0}
, the
limit
exists and equals
f
(
x
)
{\displaystyle {}f(x)}
. Because of
the Mean value theorem for definite integrals ,
for every
h
{\displaystyle {}h}
, there exists a
c
h
∈
[
x
,
x
+
h
]
{\displaystyle {}c_{h}\in [x,x+h]}
with
f
(
c
h
)
⋅
h
=
∫
x
x
+
h
f
(
t
)
d
t
,
{\displaystyle {}f(c_{h})\cdot h=\int _{x}^{x+h}f(t)dt\,,}
and therefore
f
(
c
h
)
=
∫
x
x
+
h
f
(
t
)
d
t
h
.
{\displaystyle {}f(c_{h})={\frac {\int _{x}^{x+h}f(t)dt}{h}}\,.}
For
h
→
0
{\displaystyle {}h\rightarrow 0}
,
c
h
{\displaystyle {}c_{h}}
converges to
x
{\displaystyle {}x}
, and because of the continuity of
f
{\displaystyle {}f}
, also
f
(
c
h
)
{\displaystyle {}f(c_{h})}
converges to
f
(
x
)
{\displaystyle {}f(x)}
.
◻
{\displaystyle \Box }