Chapter Contents

• Methods of Integration
• 1. Integration: The General Power Formula
• 2. Integration: The Basic Logarithmic Form
• 3. Integration: The Exponential Form
• 4. Integration: The Basic Trigonometric Forms
• 5. Integration: Other Trigonometric Forms
• 6. Integration: Inverse Trigonometric Forms
• 7. Integration by Parts
• 8. Integration by Trigonometric Substitution
• 9. Integration by Use of Tables
• Table of Common Integrals
• 10. Integration by Reduction Formulae
• 11. Integration by Partial Fractions
• Methods of Integration Problem Solver

• Comments, Questions?

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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.

Ther steps needed to decompose an algebraic fraction into its partial fractions results from a consideration of the reverse process − addition (or subtraction).

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 1

The fraction

can be expressed in partial fractions whereas the fraction

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 factored 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 2

Express the following in partial fractions.

Answer

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:

EXAMPLE 3

(a) Express the following as a sum of partial fractions.

Answer

(b) Express the following as a sum of partial fractions.

Answer

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 4

Express the following in partial fractions.

Answer

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.

EXERCISES

Write the following fractions as sum of partial fractions and then integrate with respect to x.

1.

Answer

2.

Answer