11. Integration by Partial Fractions
by M. Bourne
If the integrand (the expression after the integral sign) is in the form of an algebraic fraction and the integral cannot be evaluated by simple methods, the fraction needs to be expressed in partial fractions before integration takes place.
Consider the following addition of algebraic fractions:
In this section, we want to go the other way around. If we
start with the expression 
and
find the fractions whose sum gives this result, then the two
fractions obtained, i.e. 
and
, are called the partial fractions
of .
We decompose fractions into partial fractions like this because:
- It makes certain integrals much easier to do, and
- It is used in the Laplace transform, which we meet later.
So if we needed to integrate this fraction, we could simplify our integral in the following way:
This is now easy to integrate:
Now we will see how to split a fraction into its partial fractions.
Expressing a Fractional Function In Partial Fractions
RULE 1:
Before a fractional function can be expressed directly in partial fractions, the numerator must be of at least one degree less than the denominator.
EXAMPLE: The function 
can be
expressed in partial fractions whereas the function 
cannot be expressed directly in partial
fractions.
However, by division
and the resulting fraction can be expressed as a sum of partial fractions.
(Note: The denominator of the fraction must be factorised before you can proceed.)
RULE 2: Denominator Containing Linear Factors
For each linear factor (ax + b) in the
denominator of a rational fraction, there is a partial fraction
of the form 
, where A is a
constant.
EXAMPLE: Express 
in partial fractions.
RULE 3: Denominator Containing Repeated Linear Factors
If a linear factor is repeated n times in the denominator, there will be n corresponding partial fractions with degree 1 to n.
For example, the partial fractions for 
will be of the form:
EXAMPLES
(a) Express 
as a sum of partial
fractions.
(b) Express 
as a sum of partial
fractions.
NOTE: Scientific Notebook can do all this directly for us using Polynomials/Partial Fractions.
RULE 4: Denominator Containing a Quadratic Factor
Corresponding to any quadratic factor (ax2 +
bx + c) in the denominator, there will be a partial
fraction of the form 
.
EXAMPLE
Express 
in partial fractions.
Note: Repeated quadratic factors in the denominator are dealt with in a similar way to repeated linear factors.
Example:
Summary
| Denominator containing… | Expression | Form of Partial Fractions |
| a. Linear factor |  |
 |
| b. Repeated linear factors |  |
 |
| c. Quadratic term
(which cannot be factored) |
 |
 |
Note: In each of the above cases f(x) must be of less degree than the relevant denominator.
EXAMPLES
Write the following fractions as sum of partial fractions and then integrate with respect to x.
(a) 
(b) 
Find your integral using Mathematica!
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