Integrating Rational Functions

Integrating rational functions is a major area of A Level Maths Revision. As soon as you see the integral,

∫ (3x) / (x² − 2) dx

you might just be excited to dive in and work it out. Having a clear process is not only helpful for getting to the answer a bit faster — it will definitely help you understand better why each part works.

Noticing the Structure

First, let’s notice the structure of the integrand, which is the fraction 3x/(x² − 2). Note that the denominator is a quadratic expression and the numerator is a linear expression in x. You ought to know that in some cases the numerator in some sense is the derivative of the denominator. If you catch this sooner than later — it will save you some work as well as aid in selection of which method is more suitable.

The Idea of Substitution

The next method is even better… It definitely will be the best method. This substitution will unequivocally lead to the choice u = x² − 2. You see that du = 2x dx which is in fact a multiple of the numerator 3x dx so the integral can be written in a logarithmic form. Don’t forget to be careful about the constant factors — you will have to request the difference between 2x dx and 3x dx on separation.

When Nothing Seems to Work — Partial Fractions

When substitution does not work out precisely or the denominator can be factored into linear factors you will be using partial fractions. While x² − 2 maybe not be factored into rational linear factors over ℝ, the concept works for any rational element. Typically you would break the fraction into a summation of smaller fractions. 

Predicting the Result

Generally, by substitution you get something like ∫ (constant)(1/u) du. If you caught this early on, the integral will be of the form, ln|u|. It is important to reinsert u = x² − 2. And of course don’t forget the constant of integration, + C.

Verification and Reflection

By practising more questions like this, you will start to see patterns. You will be in a better position to determine if you use substitution or integration by parts. This alone is critical when it comes to A Level Maths Revision.

It is important to be able to break down any question into smaller steps. Seeing patterns, the correct methods, trying alternative methods, and checking results. These will help when integrating rational functions, especially complicated ones.