### Introduction

The first technique described here involves making a substitution to simplify an integral. We let a new variable equal a complicated part of the function we are trying to integrate. Choosing the correct substitution often requires experience. This skill develops with practice.

Often the technique of partial fractions can be used to write an algebraic fraction as the sum of simpler fractions. On occasions this means that we can then integrate a complicated algebraic fraction. We shall explore this approach in the second half of the section.

#### Prerequisites

- be able to find a number of simple definite and indefinite integrals
- be able to use a table of integrals
- be familiar with the technique of expressing an algebraic fraction as the sum of its partial fractions

#### Learning Outcomes

- make simple substitutions in order to find definite and indefinite integrals
- understand the technique used for evaluating integrals of the form $\int \frac{{f}^{\prime}\left(x\right)}{f\left(x\right)}dx$
- use partial fractions to express an algebraic fraction in a simpler form and integrate it

#### Contents

1 Making a substitution2 Substitution and definite integration

3 Integrals giving rise to logarithms

4 Integration using partial fractions