Math 152 Class Notes
October 6, 2015
8.4 Integration of Rational Functions by Partial Fractions
In this section we study how to integrate any rational function (a ratio of polynomials)
by expressing it as a sum of simpler fractions, called partial fractions. Let us consider
ˆ
an integral of the form
where
P (x)
and
Q(x)
P (x)
dx
Q(x)
are polynomials.
P (x) is less than the degree
P (x)/Q(x) as a sum of simpler
Q(x),
When the degree of
of
rational function
fractions.
Case 1. The denominator
Example 1.
x+5
(x − 1)(x + 2)
Q(x)
it's possible to express the
is a product of distinct linear factors.
ˆ
Example 2.
1
dx
x2 − 4
ˆ
Example 3.
x−3
dx
x3 + 6x2 + 5x
Case 2.
The denominator
Q(x)
repeated.
ˆ
Example 4.
4x
dx
(x − 1)2 (x + 1)
is a product of linear factors, some of which are
Case 3. The denominator
Q(x)
is repeated.
contains irreducible quadratic factors, none of which
ˆ
Fact used often in this section:
ˆ
Example 5.
3x2 − 4x + 5
dx
(x − 1)(x2 + 1)
x
1
1
dx = arctan
+ C.
x 2 + a2
a
a
ˆ
Example 6.
x+6
dx
(x2 + 1)(x2 + 4)
Case 4. The denominator
Q(x)
contains a repeated irreducible quadratic factor.
Example 7. Write out the form of the partial fraction decomposition of the function
1+x
(x − 1)(x2 + 4)2
Example 8. Write out the form of the partial fraction decomposition of the function
x3 + x2 + 1
x(x − 1)2 (x2 + x + 1)(x2 + 4)3
P (x) is greater or equal to the degree of the denominator Q(x), we must divide P (x) by Q(x) (by long division) until a reminder
R(x) is obtained such that deg R(x) < deg Q(x).
ˆ
x3
Example 9.
dx
x−1
When the degree of the numerator
ˆ
Example 10.
x2 + 1
dx
x2 − x