Section 8.5
Partial Fractions
In algebra you learned how to take the expression
2
1
−1
+
and simplify it to 2
.
x−1 x+ 1
x −1
In this section we’re going to learn how to decompose a rational expression back into its partial fractions in order
to make our integrals easier to integrate.
**In order to decompose a rational expression the degree of the numerator needs to be lower than the degree of
the denominator; if it is not, perform long division.
**Factor and cancel out any common factors that appear in the numerator and denominator.
Partial Fraction Decomposition Forms:
2x
A
B
A
=
+
For Linear Factors, use
.
Example:
x−r
( x+1 ) ( x −3) x +1 x−3
A
For Repeating Linear Factors, use x−r +
B
Z
+…+
.
2
( x−r )
( x−r )n
Example:
4
A
B
C
=
+
+
3
2
( x+1 ) x+ 1 ( x+1 ) ( x +1 )3
For Irreducible Quadratic Functions, use
3 x−1
( x −5 ) ( x +5 x +2 )
2
2
=
Ax + B
Cx+ D
+ 2
2
x −5 x + 5 x +2
Ax+ B
.
a x 2+bx +c
Example: