---
title: "Partial Fractions for Rational Integrands"
description: "Split a rational function into simpler fractions whose integrals are logarithms and arctangents, after checking the degree of the numerator first."
canonical: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c
source: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c.md
retrieved: 2026-09-12
---

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# Partial Fractions for Rational Integrands

Split a rational function into simpler fractions whose integrals are logarithms and arctangents, after checking the degree of the numerator first.

Subject: Mathematics · Area: Calculus & Analysis · Ages 19 to 20
Page: https://lightmysky.com/learn/mathematics/partial-fractions-for-rational-integrands-mt_PA8qxs424c

## Ready when they can

- Divide first when the numerator has degree at least that of the denominator
- Set up the correct form for a repeated linear factor
- Integrate the resulting pieces and combine the logarithms

## Lesson: Split fractions before you integrate

Partial fractions only start on a proper fraction, where the top degree is below the bottom degree. When the top is heavy, divide first: (x^2 + 3x + 5)/(x + 1) becomes x + 2 plus 3/(x + 1). Only the proper remainder gets split.

**Example.** Take 1/(x^2 - 1). Factor the bottom into (x - 1)(x + 1) and write A/(x - 1) + B/(x + 1). Strategic substitution finds them: x = 1 gives A = 1/2, and x = -1 gives B = -1/2. Each piece then integrates to a log: (1/2) ln|x - 1| - (1/2) ln|x + 1| + C.

Repeated factors need one term per power. For 5/(x + 2)^2 write A/(x + 2) + B/(x + 2)^2, not the top power alone. The same holds for x^2(x + 3): use A/x + B/x^2 + C/(x + 3).

**Tip.** Each split piece integrates to a logarithm, or an arctan for a quadratic piece. Gather the logs at the end into one compact answer. Always check by recombining over a common denominator.

**Recap.** Divide first when the top is heavy, then split and gather logarithms.

## Practice

17 questions on this page, each with its working shown.

## Needs first

- [Polynomial Division and the Factor Theorem](https://lightmysky.com/learn/mathematics/polynomial-division-and-the-factor-theorem-mt_haL2UkGa5C)
- [Trigonometric Integrals and Trigonometric Substitution](https://lightmysky.com/learn/mathematics/trigonometric-integrals-and-trigonometric-substitution-mt_JqysS41psg)

## Opens up

- [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0)
- [Improper Integrals and Their Convergence](https://lightmysky.com/learn/mathematics/improper-integrals-and-their-convergence-mt_gvtZPFRA6Z)
- [Generating Functions](https://lightmysky.com/learn/mathematics/generating-functions-mt_WAX3O-HjRk)