# Linear Partial Fractions

Linear Partial fractions are a technique used to integrate rational functions by splitting the fraction into simpler fractions. **Partial fractions** can be used to integrate rational functions of any degree, but they are particularly useful for integrating rational functions with linear and quadratic factors in the denominator.

**Proper and Improper Fractions**

There are two types of partial fractions.

- Proper fractions
- Improper fractions

A proper fraction has a numerator of lower degree than the denominator.

For example, 3/(x+1)

An improper fraction has a numerator of equal or higher degree than the denominator.

For example, (x+1)/(x+1) or (x+2)/(x+1)

**Rules for Splitting into Partial Fractions**

- Factor the denominator completely into linear and/or quadratic factors
- Decompose the rational function into fractions with numerators of 1 and denominators matching the factors
- For repeated linear factors: use a constant numerator plus a numerator with a power matching the number of times the factor repeats
- For distinct linear factors: use constants for the numerators
- For irreducible quadratic factors: use a linear numerator of the form Ax + B

**Example**

Decompose 3x/(x-1)(x+2) into partial fractions

The denominator factors as (x-1)(x+2). Using the rules above, we get:

3x/(x-1)(x+2) = A/(x-1) + B/(x+2)

Solving for A and B gives:

A = 1, B = 2

Therefore, the partial fraction decomposition is:

3x/(x-1)(x+2) = 1/(x-1) + 2/(x+2)

Integrate,

=∫ 1/(x-1) dx + ∫ 2/(x+2) dx

∫ 3x/(x-1)(x+2) dx = ln|x-1| + 2ln|x+2| + C

Therefore, decomposition into partial fractions allows easier integration of the rational function.

**FAQs**

### What if the denominator does not factor into linear terms?

If the denominator has an irreducible quadratic factor, use a partial fraction with a numerator of Ax + B.

### What if there is a repeated irreducible quadratic factor?

Use partial fractions with numerators of Ax + B and Cx + D

### How do you determine the constants A, B, C, D?

Equate coefficients of like powers and terms after multiplying both sides by the denominator.

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