Section 7.4

Partial Fractions

Integrating a messy rational function like is hard. Breaking it into makes it easy (using Logarithms).

1

The Four Cases

Goal: Rewrite as a sum of simpler fractions that we can integrate using logarithms or arctangent.

Note: If degree of degree of , first do polynomial long division.

Case I: Distinct Linear Factors

can be factored as a product of distinct linear factors:

Setup:

Example:

Case II: Repeated Linear Factors

has a repeated linear factor

Setup (for the repeated factor):

Example:

Case III: Distinct Irreducible Quadratic Factors

contains an irreducible quadratic factor (where )

Setup (for the quadratic factor):

Note: The numerator must be linear (not just a constant).

Example:

Case IV: Repeated Irreducible Quadratic Factors

contains a repeated irreducible quadratic factor

Setup (for the repeated quadratic):

Example:

Quick Reference: What Goes in the Numerator?

Denominator FactorNumerator Form
(constant)
(linear)
2

Worked Examples

Case I: Distinct Linear Factors

Evaluate .

1. Identify Case

— two distinct linear factors → Case I

2. Setup Partial Fractions

Multiply both sides by :

3. Solve for Coefficients

Method: Strategic substitution — plug in values that make factors zero.

Let :
Let :

So:

4. Integrate Using Logarithm Rules

Rule:

Here for both terms:

Combine using:

Case II: Repeated Linear Factors

Evaluate .

1. Identify Case

Denominator is — a repeated linear factor → Case II

2. Setup Partial Fractions

Multiply both sides by :

3. Solve for Coefficients

Method: Substitution + Comparing coefficients

Let :
Compare coefficients: Expand
Coefficient of :

So:

4. Integrate

First term:

Second term: (power rule: )

Case III: Irreducible Quadratic Factor

Evaluate .

1. Identify Case

is irreducible (discriminant ) → Case III

2. Setup Partial Fractions

Note: Quadratic factor gets linear numerator .

Multiply by :

3. Solve for Coefficients

Method: Substitution + Comparing coefficients

Let :
Expand and compare:

Coefficient of :
Constant term:

So:

4. Integrate

Split the second fraction:

First:

Second: (let )

Third:

Case IV: Repeated Irreducible Quadratic

Evaluate .

1. Identify Case

— repeated irreducible quadratic → Case IV

2. Setup Partial Fractions

Multiply by :

3. Solve for Coefficients

Method: Expand and compare all coefficients

Expand:
Compare coefficients:
:
:
:
:

So:

4. Integrate

Both use substitution , :

First:

Second:

3

Integration Formulas Reference

Key Formulas for Partial Fractions

Linear Factors


Quadratic Factors


Logarithm Rules



Completing the Square

For :

5

Practice Quiz

Loading...