Ultimate Guide to Integrating Rational Functions and Partial Fractions

Introduction

In lower levels of calculus, you learned how to integrate simple fractions like $\frac{1}{x}$ , which becomes $\ln|x|$ . But what happens when the denominator is a massive cubic polynomial, or when the numerator is larger than the denominator? In HSC Mathematics Extension 2, you will learn how to systematically tear down complex Rational Functions (algebraic fractions) into manageable pieces using techniques like Partial Fractions and Completing the Square.

Executive Summary

This guide covers the key methods for integrating rational functions:

Numerator $\ge$ Denominator: Using polynomial long division (or algebraic manipulation) to reduce improper fractions.
Partial Fraction Decomposition: Breaking down complex denominators into distinct linear factors, perfect squares, or irreducible quadratics.
Completing the Square: Handling irreducible quadratic denominators by forcing them into inverse tangent ( $\tan^{-1}$ ) forms.

What is this about?

A rational function is a ratio of two polynomials, $P(x) / Q(x)$ . Integration is simple if the numerator happens to be the exact derivative of the denominator (resulting in $\ln|Q(x)|$ ), but this is rarely the case in Extension 2.

If the fraction is "top-heavy" (the degree of $P(x)$ is equal to or greater than $Q(x)$ ), you must divide it out first. If it is "bottom-heavy", you must factorize the denominator $Q(x)$ . Once factorized, you use Partial Fractions to split the massive, impossible fraction into a sum of tiny, simple fractions. These tiny fractions can then be easily integrated into logarithms or inverse tangents.

Main Content

1. The Improper Fraction Rule

Rule: You can never integrate a rational function using partial fractions if the degree (highest power) of the numerator is $\ge$ the degree of the denominator.

Solution: You must use Polynomial Long Division. For example, if you have $\int \frac{x^3}{x^2 + 1} \, dx$ : Divide $x^3$ by $x^2 + 1$ . The quotient is $x$ , and the remainder is $-x$ . Rewrite the integral as: $\int \left( x - \frac{x}{x^2+1} \right) \, dx$ . Now, you can integrate $x$ easily, and the remaining fraction can be integrated using a standard log substitution!

2. Partial Fraction Decomposition

If the denominator is factorized, you can split the fraction. There are three types of factors you need to know:

Type 1: Distinct Linear Factors $\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$ These always integrate to natural logarithms: $\ln|x-a|$ .

Type 2: Perfect Square (Repeated Linear) Factors $\frac{1}{(x-a)^2 (x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$ Notice that you must include both the single power and the squared power. The squared term integrates to a power rule: $-(x-a)^{-1}$ , NOT a logarithm.

Type 3: Irreducible Quadratic Factors If a quadratic cannot be factorized over real numbers (its discriminant $\Delta < 0$ ), it requires a linear numerator: $\frac{1}{(x-a)(x^2 + cx + d)} = \frac{A}{x-a} + \frac{Bx + C}{x^2 + cx + d}$ These integrate into a combination of a logarithm and an inverse tangent.

3. Finding the Constants ( $A, B, C$ )

Once you set up the partial fraction equation, multiply both sides by the original denominator to get a flat algebraic equation. To find the constants, you can:

The Cover-Up Method / Substitution: Substitute strategic values of $x$ that make the brackets equal zero. This immediately reveals constants like $A$ .
Equating Coefficients: Expand the right side and equate the coefficients of $x^2$ , $x$ , and the constants with the left side to form simultaneous equations.

4. Completing the Square

If you are left with an irreducible quadratic in the denominator like $\frac{1}{x^2 + 4x + 13}$ , you cannot use partial fractions on it. Instead, complete the square!

$x^2 + 4x + 13 = (x^2 + 4x + 4) + 9 = (x+2)^2 + 3^2$ . The integral $\int \frac{1}{(x+2)^2 + 3^2} \, dx$ perfectly matches the standard integral formula for inverse tangent: $\frac{1}{a} \tan^{-1}(\frac{x}{a})$ . Result: $\frac{1}{3} \tan^{-1}\left(\frac{x+2}{3}\right)$ .

Simple Worked Example

Question: Evaluate $\int \frac{5x - 1}{(x-1)(x+3)} \, dx$ .

Solution: Step 1: Set up the Partial Fractions. Because the denominator has distinct linear factors, we write: $\frac{5x - 1}{(x-1)(x+3)} = \frac{A}{x-1} + \frac{B}{x+3}$

Step 2: Clear the denominators. Multiply the entire equation by $(x-1)(x+3)$ : $5x - 1 = A(x+3) + B(x-1)$

Step 3: Solve for $A$ and $B$ (Substitution method). Let $x = 1$ (to eliminate $B$ ): $5(1) - 1 = A(1+3) + B(0)$ $4 = 4A \implies A = 1$

Let $x = -3$ (to eliminate $A$ ): $5(-3) - 1 = A(0) + B(-3 - 1)$ $-16 = -4B \implies B = 4$

Step 4: Rewrite the integral. $\int \frac{5x - 1}{(x-1)(x+3)} \, dx = \int \left( \frac{1}{x-1} + \frac{4}{x+3} \right) \, dx$

Step 5: Integrate. Both terms are simple standard forms that integrate to natural logarithms. $= \ln|x-1| + 4\ln|x+3| + C$

mini-FAQ page

Q: Can I use complex numbers to factorize an irreducible quadratic and then use partial fractions? A: Theoretically yes, but the syllabus requires you to integrate over the real numbers. You should keep it as an irreducible quadratic over the reals (using the $\frac{Bx+C}{x^2+c}$ format) or complete the square to use inverse tangent.

Q: Do I really have to use long division if the powers are equal? A: Yes! Even if it is $\frac{x}{x+1}$ , the powers are both 1. You can use algebraic manipulation instead of long division to be faster: $\frac{x+1-1}{x+1} = 1 - \frac{1}{x+1}$ . Either way, you must break it apart first.

Common mistakes to avoid

Forgetting the $Bx+C$ numerator: If the denominator has an irreducible quadratic like $(x^2+4)$ , the numerator above it MUST be $Bx+C$ . If you just write $B$ , your simultaneous equations will be impossible to solve correctly.
Forgetting the absolute value signs: When integrating $\frac{1}{x-a}$ , the answer is $\ln|x-a|$ , not $\ln(x-a)$ . The absolute value signs are strictly required because logarithms are undefined for negative numbers.

Practice on Vu's Maths Hub

Partial fractions require impeccable algebraic accuracy.

Practice long division and setting up $A, B, C$ equations in the HSC Integration Booklet.
Master completing the square with our interactive modules at Vu's Maths Hub.

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