You are here

How to Integrate It

Seán M. Stewart
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Salim Salem
, on

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:

  1. Riemann’s integral is discussed in chapters one, two, four and sixteen.
  2. Improper integrals are the topic of twentieth and twenty first chapters.
  3. 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.

1. The Riemann integral
2. Basic properties of the definite integral – Part I
3. Some basic standard forms
4. Basic properties of the definite integral – Part II
5. Standard forms
6. Integration by substitution
7. Integration by parts
8. Trigonometric integrals
9. Hyperbolic integrals
10. Trigonometric and hyperbolic substitutions
11. Integrating rational functions by partial fraction decomposition
12. Six useful integrals
13. Inverse hyperbolic functions and integrals leading to them
14. Tangent half-angle substitution
15. Further trigonometric integrals
16. Further properties for definite integrals
17. Integrating inverse functions
18. Reduction formulae
19. Some other special techniques and substitutions
20. Improper integrals
21. Two important improper integrals
Appendix A. Partial fractions
Appendix B. Answers to selected exercises