Calculus II
Stewart
Dr. Berg
Spring 2010
Page 1
8.4
8.4
Integration by Partial Fractions
We can integrate rational functions by writing them as sums of simpler fractions.
Example A
Since
1
x
−
1
+
1
x
2
+
1
=
x
2
+
1
( )
+
x
−
1
( )
x
−
1
( )
x
2
+
1
( )
=
x
2
+
x
x
3
−
x
2
+
x
−
1
, then
x
2
+
x
x
3
−
x
2
+
x
−
1
∫
dx
=
1
x
−
1
+
1
x
2
+
1
∫
dx
=
ln
x
−
1
+
arctan
x
+
C
.
Basic Idea
Given a rational function
P
(
x
)
Q
(
x
)
, we begin by completely factoring the
denominator
Q
(
x
)
and then write
P
(
x
)
Q
(
x
)
as a sum of rational functions (partial fractions)
whose denominators are (powers of) the factors of
Q
(
x
)
.
We encounter four cases: linear
factors of multiplicity one, linear factors of higher multiplicity, irreducible quadratic
factors of multiplicity one, and irreducible quadratic factors of higher multiplicity.
The first step is to use long division to rewrite the rational function as a
polynomial plus a proper rational function (the degree of the numerator is smaller than
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 Spring '09
 GOGOLEV
 Calculus, Fractions, Rational Functions, Fraction, Rational function, dx, Dr. Berg, proper rational function

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