21. Indefinite Integrals
P. K. Lamm
11/02/11 (15:10)
p. 1 / 7
Lecture Notes:
21. Indefinite Integrals
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of definitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. Antiderivatives
Definition:
A function
F
(
x
) is called an
antiderivative
of the function
f
(
x
) if
F
satisfies
F
0
(
x
) =
f
(
x
)
for all
x
in the domain of
f
.
Example 1.1:
We know
(sin
x
)
0
= cos
x,
all
x,
so
F
(
x
) = sin
x
is an antiderivative of
f
(
x
) = cos
x
.
Example 1.2:
We know
(3
x
2
+
x
)
0
= 6
x
+ 1
,
all
x,
so
F
(
x
) = 3
x
2
+
x
is an antiderivative of
f
(
x
) = 6
x
+ 1.
Also note that
(3
x
2
+
x
+ 5)
0
= 6
x
+ 1
,
all
x,
so
F
(
x
) = 3
x
2
+
x
+ 5 is also an antiderivative of
f
(
x
) = 6
x
+ 1. In fact, for any constant
C
,
F
(
x
) = 3
x
2
+
x
+
C
is an antiderivative of
f
(
x
) = 6
x
+ 1 because
(3
x
2
+
x
+
C
)
0
= 6
x
+ 1
,
all
x.
From the Mean Value Theorem we have the following:
Theorem:
If
F
(
x
) is an antiderivative of
f
(
x
), then
all
antiderivatives of
f
(
x
) are given by
G
(
x
) =
F
(
x
) +
C,
for
C
an arbitrary constant.
Then for
F
(
x
) any antiderivative of
f
(
x
), the function
F
(
x
) +
C
gives the
most general
formula
for
all
antiderivatives of
f
(
x
).
Definition:
The set of all antiderivatives of
f
is called the
indefinite integral of
f
, written
Z
f
(
x
)
dx.
1
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21. Indefinite Integrals
P. K. Lamm
11/02/11 (15:10)
p. 2 / 7
Thus if
F
(
x
) is
any
antiderivative of
f
,
Z
f
(
x
)
dx
=
F
(
x
) +
C,
for
C
an arbitrary constant.
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 Fall '10
 KIHYUNHYUN
 Calculus, Derivative, Fundamental Theorem Of Calculus, Integrals, 1 sec, P. K. Lamm

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