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Section 8.4:
Integration of Rational Functions by Partial Fractions
This technique of integration allows us to express a rational function (which is a quotient
of polynomials) as a sum of simpler fractions (called partial fractions), which are more
easily integrated than the original rational function.
In other words, we are faced with
f x
(29
dx
∫
, where
f x
( 29
=
P
(
x
)
Q
(
x
)
is a rational function.
Let
deg(
P
)
be the degree of the polynomial in the numerator and
deg(
Q
)
be the degree of
the polynomial in the denominator.
Note that if
deg(
P
)
<
deg(
Q
)
, then
f
is a proper rational function; and if
deg(
P
)
≥
deg(
Q
)
,
then
f
is an improper rational function.
Procedure to integrate a rational function using partial fractions:
Step 1.
Determine if
f
is a proper or an improper rational function.
(i)
If
f
is proper, then
f x
( 29
=
P
(
x
)
Q
(
x
)
is the rational function that will be
integrated using partial fractions.
Go to Step 2.
(ii)
If
f
is improper, perform long division of polynomials to obtain
f x
(29
=
Sx
(29
+
Rx
(29
Qx
(29
(where
R
is the remainder), and
R x
( 29
Q x
( 29
is the rational
function that will be integration using partial fractions.
Go to Step 2.
Step 2.
Factor
Qx
( 29
as far as possible.
(The factored from of
Qx
( 29
will contain linear
and/or irreducible quadratic factors.
An irreducible quadratic factor is a factor of the
form
ax
2
+
bx
+
c
for which
b
2

4
ac
<
0
.)
Step 3.
Perform the Partial Fraction Decomposition on the rational function.
(i)
If
x

r
is a distinct linear factor appearing in the factored form of
Qx
( 29
,
then the partial fraction decomposition will contain a term of the form
A
1
x

r
(where
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This note was uploaded on 01/12/2010 for the course MATH 44245 taught by Professor Famiglietti during the Fall '07 term at UC Irvine.
 Fall '07
 FAMIGLIETTI
 Math, Polynomials, Fractions, Rational Functions

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