1. Integration of Rational Functions
7.4
Integration of Rational Functions
By Partial Fractions

2. Example 1 This would be a lot easier if we could
re-write it as two separate terms.
Multiply by the common denominator.
Set like-terms equal to each other.
Solve two equations with two unknowns.

3. This technique is called
Partial Fractions
Solve two equations with two unknowns.

4. The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside.
Multiply by the common denominator.
Let x = - 1
Let x = 3

5. Method of Partial Fractions
If the fraction is improper, use long division to rewrite it as a sum of a polynomial and a proper fraction.
If the fraction is proper, factor the denominator completely and write it as a sum of fractions as follows:
a) For each linear factors (ax+b)n, the decomposition must have the form:
b) ) For each irreducible quadratic factors (ax2+bx+c)n, the decomposition must have the form:

6. Example 2
Repeated roots: we must use two terms for partial fractions.

7. Example 3
If the degree of the numerator is higher than the degree of the denominator, use long division first.
(from example one)

8. Example 4
irreducible
quadratic
factor
repeated root

10. Examples