du
dx
(by the Chain Rule)
=
e
u
2
du
dx
(by FTCI)
=
e
x
4
·
2
x
which is the same answer we calculated by the previous method.
In the statement of the Fundamental Theorem of Calculus, the lower limit of the
integral was always fixed. That is, it did not vary with
x
. We can now make our
example even more complicated by letting the lower limit of the integral vary as a
function of
x
. Let
H
(
x
)
=
Z
x
2
cos(
x
)
e
t
2
dt
.
How would we find
H
0
(
x
)?
We can cleverly use the properties of the integral. In fact, we can write
H
(
x
)
=
Z
x
2
cos(
x
)
e
t
2
dt
=
Z
3
cos(
x
)
e
t
2
dt
+
Z
x
2
3
e
t
2
dt
.
Furthermore, we know that
Z
3
cos(
x
)
e
t
2
dt
=

Z
cos(
x
)
3
e
t
2
dt
and this integral is in the form where we can use the Fundamental Theorem.
Therefore, we have that
H
(
x
)
=
Z
x
2
cos(
x
)
e
t
2
dt
=
Z
x
2
3
e
t
2
dt

Z
cos(
x
)
3
e
t
2
dt
.
If we now let
H
1
(
x
)
=
Z
cos(
x
)
3
e
t
2
dt
then
H
(
x
)
=
G
(
x
)

H
1
(
x
)
where
G
(
x
) is defined as before. But then
H
0
(
x
)
=
G
0
(
x
)

H
1
0
(
x
)
Calculus 2
(B. Forrest)
2
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Chapter 1: Integration
40
and we already know that
G
0
(
x
)
=
2
xe
x
4
.
This means that we only need to find
H
1
0
(
x
). To accomplish this we do exactly what
we did to find
G
0
(
x
). We note that
H
1
(
x
)
=
F
(cos(
x
))
so
H
1
0
(
x
)
=
F
0
(cos(
x
))
d
dx
(cos(
x
))
=

sin(
x
)
e
(cos(
x
))
2
Combining all of this together gives us that
H
0
(
x
)
=
G
0
(
x
)

H
1
0
(
x
)
=
2
xe
x
4

(

sin(
x
)
e
(cos(
x
))
2
)
=
2
xe
x
4
+
sin(
x
)
e
(cos(
x
))
2
The previous example leads us to an extended version of the Fundamental Theorem
of Calculus.
THEOREM 6
Extended Version of the Fundamental Theorem of Calculus
Assume that
f
is continuous and that
g
and
h
are di
↵
erentiable. Let
H
(
x
)
=
Z
h
(
x
)
g
(
x
)
f
(
t
)
dt
.
Then
H
(
x
) is di
↵
erentiable and
H
0
(
x
)
=
f
(
h
(
x
))
h
0
(
x
)

f
(
g
(
x
))
g
0
(
x
).
1.6
The Fundamental Theorem of Calculus (Part 2)
We have seen that the Fundamental Theorem of Calculus provides us with a
simple
rule for di
↵
erentiating integral functions
and so it provides the key link between
di
↵
erential and integral calculus. However, we will soon see it also provides us with
a
powerful tool for evaluating integrals
. First we must briefly review the topic of
antiderivatives from your study of di
↵
erential calculus.
Calculus 2
(B. Forrest)
2
Section 1.6: The Fundamental Theorem of Calculus (Part 2)
41
1.6.1
Antiderivatives
We know a number of techniques for calculating derivatives. In this section, we will
review how we can sometimes “undo” di
↵
erentiation. That is, given a function
f
,
we will look for a new function
F
with the property that
F
0
(
x
)
=
f
(
x
).
DEFINITION
Antiderivative
Given a function
f
, an
antiderivative
is a function
F
such that
F
0
(
x
)
=
f
(
x
)
.
If
F
0
(
x
)
=
f
(
x
) for all
x
in an interval
I
, we say that
F
is an antiderivative for
f
on
I
.
EXAMPLE 12
Let
f
(
x
)
=
x
3
. Let
F
(
x
)
=
x
4
4
. Then
F
0
(
x
)
=
4
x
4

1
4
=
x
3
=
f
(
x
)
,
so
F
(
x
)
=
x
4
4
is an antiderivative of
f
(
x
)
=
x
3
.
While the derivative of a function is always unique, this is
not
true of antiderivatives.