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The Fundamental Theorem of Calculus (Part1)
If f is continuous on an interval I, then f has an antiderivative on I.
In particular, if a is any number in I, then the function F defined by
∫
=
x
a
dt
t
f
x
F
)
(
)
(
is an antiderivative of f on I; that is, F’(x) = f(x) for each x in I, or in
an alternative notation
∫
=
x
a
x
f
dt
t
f
dx
d
)
(
]
)
(
[
Proof:
If
F(x)
is defined for all
x
in the interval
I
, then we will prove that
F(x)
is defined at each
x
in the interval
I
.
F’(x) =
x
w
x
F
w
F
x
w


→
)
(
)
(
lim
Since
F’(x) = f(x)
for each value x,
F(w) =
∫
w
a
dt
t
f
)
(
and
F(x) =
∫
x
a
dt
t
f
)
(
=
∫
∫


→
w
a
x
a
x
w
dt
t
f
dt
t
f
x
w
])
)
(
)
(
[
1
(
lim
By the properties of definite integrals, the integral from
a
to
x
is equal to the negative integral from
x
to
a
.
=
∫
∫
+

→
w
a
a
x
x
w
dt
t
f
dt
t
f
x
w
])
)
(
)
(
[
1
(
lim
By the other properties of definite integrals, the integral from
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