MATH1231 - Integrating Rational Functions

A rational function is defined as

$f(x) = \frac{p(x)}{q(x)}$

where p and q are polynomials.

Every rational function has a primitive among the elementary functions.

Proper vs Improper

F is ‘proper’ if the degree of q is greater than the degree of p.

Else it is ‘improper’.


F is irreducible if it has no real linear factors (e.g. with $ax^2 + bx + c$, it is irreducible if its discriminant ($b^2 - 4ac$) is negative).

Simpler Tactics

Make $p(x)$ be the derivative of $q(x)$, and then integrate to $ln(qx)$

New Approach:

If it is improper, use polynomial division to write f as the sum of a polynomial and a proper rational fractional.

Proper rational functions can be written as a unique sum of functions of the form:

\begin{align} \frac {A}{(x-a)^k} and \frac {Bx + C}{(x^2 + bx + c)^k} \end{align}

Where the quadratic is irreducible.

This sum is called the partial fractions decompositions of f.

Once the is in that form, we need only complete the square, use a substitution or manipulate it algebraically before integrating it normally.