Partial Fractions
Math 114, Fall 2010
Learning Objectives
•
To be able to evaluate integrals using partial fractions.
•
To be able to rewrite rational functions as the sum of functions by the process of
partial fraction
decomposition
.
Example,
Z
2
1

x
2
dx
=
Z
dx
1 +
x
+
dx
1

x
= ln

1 +
x
 
ln

1

x

+
C.
This is called a
partial fraction decomposition
.
Writing Rational Functions as Partial Fractions
To see how the method of partial fractions works in general, consider the rational function,
f
(
x
) =
P
(
x
)
Q
(
x
)
where
P
and
Q
are polynomials. It is possible to express f as a sum of simpler fractions provided that the
degree of
P
is less than the degree of
Q
. For example, to find the partial fractions decomposition for
f
(
x
) =
6
x
2

1
begin by factoring the denominator:
6
x
2

1
=
6
(
x

1)(
x
+ 1)
Now assume that there are constants
A
and
B
so that
6
(
x

1)(
x
+ 1)
=
A
x

1
+
B
x
+ 1
Add the two fractions together by finding a common denominator
A
x

1
+
B
x
+ 1
=
A
x

1
·
(
x
+ 1)
(
x
+ 1)
+
B
x
+ 1
·
(
x

1)
(
x

1)
6
(
x

1)(
x
+ 1)
=
A
(
x
+ 1)
(
x

1)(
x
+ 1)
+
B
(
x

1)
(
x

1)(
x
+ 1)
=
A
(
x
+ 1) +
B
(
x

1)
(
x

1)(
x
+ 1)
1
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For two polynomials to be equivalent, the corresponding coefficients must be equal. Since there is no
x
term
in the original numerator, the coefficient of
x
must be 0. The constant term in the original numerator is 6,
the constant term in the expression on the right must be 6 as well.
6 =
A
(
x
+ 1) +
B
(
x

1)
6 =
Ax
+
A
+
Bx

B
The following two equations can be solved simultaneously to find A and B
6 =
A

B
0 =
A
+
B
The solution to these equations are
A
= 3
and
B
=

3
, so the equation can be rewritten as:
6
x
2

1
=
3
x

1

3
x
+ 1
Integrating Rational Functions with Partial Fractions
If we want to integrate the function
Z
6
x
2

1
dx
we can integrate the partial fractions
Z
6
x
2

1
dx
=
Z
3
x

1

3
x
+ 1
dx
=
Z
3
x

1
dx

Z
3
x
+ 1
dx
Integrating each function, we get:
3 ln

x

1
 
3 ln

x
+ 1

+
C
Other Situations
1. If the degree (highest power) of P(x) is equal to or greater than the degree of Q(x), then you must
use polynomial division in order to rewrite the given rational function as the sum of a polynomial and
a new rational function. For example, in the rational function below, the degree of the numerator is
greater than the degree of the denominator. We can use polynomial division to rewrite the function so
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 Spring '11
 PartialFractions
 Calculus, Fractions, Integrals, Rational Functions, Fraction, Elementary algebra, Rational function, dx, Partial fractions in integration, ln x

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