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Math 1314
Day 7
Section 14.1  Antiderivatives
So far in this course, we have been interested in finding derivatives and in the
applications of derivatives.
In this chapter, we will look at the reverse process.
Here we
will be given the “answer” and we’ll have to find the “problem.”
This process is generally called
integration
.
We can use integration to solve a variety of
problems.
Antiderivatives
Definition
:
A function
F
is an antiderivative of
f
on interval
I
if
( )
( )
F x
f x
′
=
for all
x
in
I.
The process of finding an antiderivative is called
antidifferentiation
or
finding an
indefinite integral
.
Example
1
:
Determine if
F
is an antiderivative of
f
if
5
2
2
3
3
1
)
(
2
3
+
+
+
=
x
x
x
x
F
and
.
2
3
)
(
2
+
+
=
x
x
x
f
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View Full Document Notation
:
We will use the integral sign
∫
to indicate integration (antidifferentiation).
Problems will be written in the form
∫
+
=
.
)
(
)
(
C
x
F
dx
x
f
This indicates that the
indefinite integral of
)
(
x
f
with respect to the variable
x
is
C
x
F
+
)
(
where
)
(
x
F
is an
antiderivative of
f.
Basic Rules
Rule 1:
The Indefinite Integral of a Constant
∫
+
=
C
kx
dx
k
Example 3
:
∫
dx
5
Rule 2:
The Power Rule
1
,
1
1

≠
+
+
=
∫
+
n
C
n
x
dx
x
n
n
Example 4
:
∫
dx
x
4
Example 5
:
4
x dx
∫
Example 6
:
3
1
dx
x
∫
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View Full Document Rule 3:
The Indefinite Integral of a Constant Multiple of a Function
∫
∫
=
dx
x
f
c
dx
x
cf
)
(
)
(
Example 7
:
∫
dx
x
3
4
Example 8
:
4
2
dx
x
∫
Rule 4:
The Sum (Difference) Rule
[ ]
∫
∫
∫
±
=
±
dx
x
g
dx
x
f
dx
x
g
x
f
)
(
)
(
)
(
)
(
Example 9
:
∫
+
+
dx
x
x
)
1
5
2
(
2
Rule 5:
The Indefinite Integral of the Exponential Function
∫
+
=
C
e
dx
e
x
x
Example 10
:
∫

dx
x
e
x
)
4
5
(
3
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This note was uploaded on 02/21/2012 for the course MATH 1314 taught by Professor Marks during the Fall '08 term at University of Houston.
 Fall '08
 MARKS
 Antiderivatives, Derivative

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