# Partial fractions

Split a fraction with a factorised denominator into simpler fractions that are easy to integrate.

Maths (university) · Part 06: Integration techniques and uses · Card 3 of 9 · Last updated 2026-10-19

## Analogy

Breaking a large order into separate small parcels that each fit through the door.

## Example

**Integral of 1/((x-1)(x+1)) dx**

Write as (1/2)[1/(x-1) - 1/(x+1)].
Check at x = 2: 1/3 = (1/2)(1 - 1/3).
Integrate each part.
Result: (1/2) ln|(x-1)/(x+1)| + C.

## In practice

Used for rational functions, which are polynomial over polynomial.

## Practice

### Flashcards

- **How do you split 1/((x-1)(x+1))?** (1/2)[1/(x-1) - 1/(x+1)].

## Questions people ask

### How do I find the numbers in a partial fraction split?

Write the split with unknown constants, such as A over x minus 1 plus B over x plus 1. Multiply both sides by the common denominator, then substitute values that make terms vanish. Setting x = 1 gives A directly, and x = -1 gives B.

### When do I use partial fractions?

Use them when the integrand is a polynomial divided by a polynomial that factorises, and substitution does not work. If the top has degree at least that of the bottom, divide first, then split the remainder. Long division comes first, because partial fractions only works on a proper fraction.

## Related concepts

- [Integration by parts](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-1)
- [Choosing u and dv](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-2)
- [Trigonometric integrals](https://studyvisually.com/learn/science/maths-university/maths-university-part-6/maths-university-part-6-concept-4)
- [Integration by substitution](https://studyvisually.com/learn/science/maths-university/maths-university-part-5/maths-university-part-5-concept-7)

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