*How to Integrate It* is meant to be a practical guide to finding primitives of continuous functions; though it sometimes ventures a little outside this scope.

The book can be divided into three essential parts:

- Riemann’s integral is discussed in chapters one, two, four and sixteen.
- Improper integrals are the topic of twentieth and twenty first chapters.
- Integration techniques fill the remaining chapters: standard forms (Chapters 3 and 5), substitution (chapters 6 and 19) integration by parts (chapter 7), trigonometric integrals (chapters 8, 10, 14 and 15), hyperbolic integrals (chapters 9, 10 and 13), integrating rational functions (chapter 11 and appendix A). Two other chapters deal with integrals of inverse functions (chapter 17) and reduction formulae (chapter 18). Appendix B gives the answers to some selected exercises.

Each chapter of this book starts with a quote, then a little motivating introduction or example, followed by a definition, a rule or some properties, then a wealth of practical examples and exercises which range in difficulty. The author does not attempt to prove any deep theorems. Even the basic properties of Riemann’s integral (chapter 2) are mostly admitted, due to the inherent difficulties in working with Darboux’s definition.

The chapters concerned with the integration techniques are finely written: they are short with minimal theoretical explanation, good practical rules, and a great number of examples and exercises. Students and teachers can find a lot of interesting things to learn or use.

The last two chapters are meant to show students some important applications of integrals but Stewart chooses difficult applications that are demanding. Students will need to know limits of sequences and series and deeper properties of integrals to fully understand and be able to work with improper integrals.

This book is a very good introduction to the techniques of integration. It is not a theoretical book on integration; indeed, most of it can be well understood by pre-university students who are learning integral calculus. Yet Darboux’s definition, the improper integrals and the structure of the book may hinder students’ ability to fully grasp integration which, as Stewart says, “is hard. Very hard”.

Salim Salem is Professor of Mathematics at the Saint-Joseph University of Beirut.