Partial Fractions and Their Forms

This lesson introduces partial fractions, the tool we use to rewrite a rational function as a sum of simpler fractions before integrating. It explains when a rational function is proper and then sets out the six standard forms you will use, depending on the factors in the denominator.

Proper rational functions

A rational function $\tfrac{p(x)}{q(x)}$, with $q(x)\neq 0$, is called proper when the degree of the numerator is less than the degree of the denominator. If the denominator $q(x)$ factorises into linear factors and quadratic factors, then to integrate $\tfrac{p(x)}{q(x)}$ we first write it as a sum of two or more simpler rational functions. This splitting is the partial fraction method, and there are six standard forms.

Form 1: distinct linear factors

When the denominator is a product of two distinct linear factors, the fraction splits into one term per factor:

$\frac{px+q}{(x\pm a)(x\pm b)} = \frac{A}{x\pm a} + \frac{B}{x\pm b}, \quad a\neq b$

Here $a\neq b$, and each $\pm$ sign is copied exactly as it appears in the question.

Form 2: a repeated linear factor

When a linear factor is squared, you need one term for the factor and one for its square:

$\frac{px+q}{(x\pm a)^{2}} = \frac{A}{x\pm a} + \frac{B}{(x\pm a)^{2}}$

Form 3: three distinct linear factors

A quadratic numerator over three distinct linear factors splits into three terms, one per factor. A linear factor is one where the power of $x$ is $1$.

$\frac{px^{2}+qx+r}{(x\pm a)(x\pm b)(x\pm c)} = \frac{A}{x\pm a} + \frac{B}{x\pm b} + \frac{C}{x\pm c}$

Form 4: a repeated factor with another linear factor

When one linear factor is squared and another is distinct, the squared factor contributes two terms:

$\frac{px^{2}+qx+r}{(x\pm a)^{2}(x\pm b)} = \frac{A}{x\pm a} + \frac{B}{(x\pm a)^{2}} + \frac{C}{x\pm b}$

Form 5: an irreducible quadratic factor

When the denominator contains a quadratic that cannot be factorised, that factor gets a linear numerator $Bx+C$:

$\frac{px^{2}+qx+r}{(x\pm a)(x^{2}+bx+c)} = \frac{A}{x\pm a} + \frac{Bx+C}{x^{2}+bx+c}$

Here $x^{2}+bx+c$ is a quadratic that cannot be factorised.

Form 6: an improper fraction

If the degree of the numerator is greater than or equal to the degree of the denominator, first divide and write the fraction as

$\text{quotient} + \frac{\text{remainder}}{\text{divisor}}.$

Then integrate, applying the partial fraction method to the second piece, $\tfrac{\text{remainder}}{\text{divisor}}$. This step of dividing first is an important one to remember.

A note on the numerator

In all of these cases, copy each $\pm$ sign exactly as it appears in the question. The numerator is not always of the form $px+q$: it may be a single $x$, a full expression $px+q$, the constant $1$, or any other constant. The next lesson works through questions using these forms.

Key takeaways

• A rational function is proper when the numerator's degree is less than the denominator's; only then do you split it directly.
• Each distinct linear factor gives one term; a repeated linear factor gives a term for each power; an irreducible quadratic factor gets a linear numerator $Bx+C$.
• If the fraction is improper, divide first to get a quotient plus a proper remainder, then apply partial fractions to the remainder.