Decompose rational functions into partial fractions and use the decomposition to integrate them.

What this dot point is asking

You need to decompose rational functions into partial fractions and use the decomposition to integrate them.

When and how to decompose

Partial fractions apply to a proper rational function (numerator degree less than denominator degree). If the function is improper, do polynomial division first and decompose the remaining proper part.

Finding the constants

Two reliable methods: substitute strategic $x$-values (the cover-up idea), or expand and equate coefficients of like powers.

Integrating the pieces

Each $\dfrac{A}{x - a}$ integrates to $A\ln|x - a|$, using $\int \dfrac{1}{x - a}\,dx = \ln|x - a| + C$. So once decomposed, the integral is a sum of logarithms.

Improper rational functions

If the numerator degree is at least the denominator degree, divide first. For instance $\dfrac{x^2}{x^2 - 1}$ is improper; dividing gives $1 + \dfrac{1}{x^2 - 1}$, and only the proper remainder $\dfrac{1}{(x-1)(x+1)}$ is decomposed.