Integration by Partial Fractions

7.5 Integration by Partial Fractions

Understanding Rational Functions

A rational function is the ratio of two polynomials:

where $P(x)$ and $Q(x)$ are polynomials in $x$, and $Q(x) \neq 0$.

Proper vs Improper Rational Functions

• Proper rational function: degree of $P(x)$ is less than degree of $Q(x)$
• Improper rational function: degree of $P(x)$ is greater than or equal to degree of $Q(x)$

Handling Improper Rational Functions

If $\frac{P(x)}{Q(x)}$ is improper, we perform long division to write:

where $T(x)$ is the polynomial quotient and $\frac{P_1(x)}{Q(x)}$ is a proper rational function (the remainder). Since polynomials are easy to integrate, the problem reduces to integrating a proper rational function.

The Method of Partial Fractions

We consider rational functions whose denominators factorise into linear factors (of the form $x-a$) and irreducible quadratic factors (of the form $x^2+bx+c$). Any proper rational function $\frac{P(x)}{Q(x)}$ can be expressed as a sum of simpler rational functions — its partial fraction decomposition — and each term is then integrated by known methods.

Table of Partial Fraction Forms

where $x^2+bx+c$ cannot be factorised further, and $A$, $B$, $C$ are real numbers to be determined.

• For distinct linear factors (Type 1 and 3): each factor contributes one term with a constant numerator.
• For repeated linear factors (Type 2 and 4): a factor $(x-a)^k$ contributes $k$ terms: $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_k}{(x-a)^k}$.
• For irreducible quadratic factors (Type 5): the numerator is linear ($Bx+C$), not constant.

Integration Formulas Used

General Strategy

1. Check if proper: if degree of numerator ≥ degree of denominator, perform long division first. …