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ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c
°
W W L Chen, X T Duong and Macquarie University, 1999.
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Chapter 14
INTRODUCTION TO INTEGRATION
14.1. Antiderivatives
In this chapter, we discuss the inverse process of diFerentiation. In other words, given a function
f
(
x
), we
wish to ±nd a function
F
(
x
) such that
F
0
(
x
)=
f
(
x
). Any such function
F
(
x
) is called an antiderivative,
or inde±nite integral, of the function
f
(
x
), and we write
F
(
x
Z
f
(
x
)d
x.
A ±rst observation is that the antiderivative, if it exists, is not unique. Suppose that the function
F
(
x
) is an antiderivative of the function
f
(
x
), so that
F
0
(
x
f
(
x
). Let
G
(
x
F
(
x
)+
C
, where
C
is any ±xed real number. Then it is easy to see that
G
0
(
x
F
0
(
x
f
(
x
), so that
G
(
x
) is also an
antiderivative of
f
(
x
). A second observation, somewhat less obvious, is that for any given function
f
(
x
),
any two distinct antiderivatives of
f
(
x
) must diFer only by a constant. In other words, if
F
(
x
) and
G
(
x
)
are both antiderivatives of
f
(
x
), then
F
(
x
)
−
G
(
x
) is a constant. In this chapter, we shall denote any
such constant by
C
, with or without subscripts.
An immediate consequence of this second observation is the following simple result related to the
derivatives of constants in Section 11.1.
ANTIDERIVATIVES OF ZERO.
We have
Z
0d
x
=
C.
In other words, the antiderivatives of the zero function are precisely all the constant functions.
Indeed, many antiderivatives can be obtained simply by referring to various rules concerning deriva
tives. We list here a number of such results. The ±rst of these is related to the constant multiple rule
for diFerentiation in Section 11.2.
†
This chapter was written at Macquarie University in 1999.
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W W L Chen and X T Duong : Elementary Mathematics
CONSTANT MULTIPLE RULE.
Suppose that a function
f
(
x
)
has antiderivatives. Then for any
Fxed real number
c
,wehave
Z
cf
(
x
)d
x
=
c
Z
f
(
x
x.
ANTIDERIVATIVES OF POWERS.
(a) Suppose that
n
is a Fxed real number such that
n
6
=
−
1
. Then
Z
x
n
d
x
=
1
n
+1
x
n
+1
+
C.
(b) We have
Z
x
−
1
d
x
= log

x

+
C.
Proof.
Part (a) is a consequence of the rule concerning derivatives of powers in Section 11.1. If
x>
0,
then part (b) is a consequence of the rule concerning the derivative of the logarithmic function in Section
12.3. If
x<
0, we can write

x

=
u
, where
u
=
−
x
. It then follows from the Chain rule that
d
d
x
(log

x

)=
d
u
d
x
×
d
d
u
(log
u
−
1
u
=
1
x
(1)
again.
♣
Corresponding to the sum rule for diFerentiation in Section 11.2, we have the following.
SUM RULE.
Suppose that functions
f
(
x
)
and
g
(
x
)
have antiderivatives. Then
Z
(
f
(
x
)+
g
(
x
))d
x
=
Z
f
(
x
x
+
Z
g
(
x
x.
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