Concept preview

What you will be able to do

1. 01Recognise when a rational integrand is PROPER (numerator degree strictly less than denominator degree) and, when it is not, use polynomial long division first.
2. 02Decompose a proper rational function with distinct linear factors, $\dfrac{p(x)}{(x-a)(x-b)} = \dfrac{A}{x-a}+\dfrac{B}{x-b}$, and find $A$ and $B$ efficiently with the cover-up (Heaviside) method.
3. 03Integrate the resulting simple pieces to a sum of logarithms, and evaluate the corresponding definite integrals.

A rational function is PROPER when the degree of the numerator is strictly less than the degree of the denominator. Partial fractions applies directly only to proper fractions, so this is the first thing to check. If the fraction is improper (numerator degree $\ge$ denominator degree), you must do polynomial long division first to peel off a polynomial part, leaving a proper remainder fraction to decompose. For now assume we have a proper fraction whose denominator factors into two DISTINCT linear pieces, $(x-a)(x-b)$.