Most high school students think that all of mathematics is building up towards the final goal of calculus, and after that crescendo is reached all of mathematics is just doing calculus over and over again. I'm sure we have all had a student who has asked us what research in mathematics is and then expressed surprise when we tell them it is not just computing harder and harder integrals. These students would have their ideas (or fears?) validated if they picked up the new book Irresistible Integrals, by George Boros and Victor H. Moll. In this book, the authors write about many areas of mathematics which come up during the evaluation of integrals.
The authors claim — and for the most part correctly — to assume only the topics covered in a calculus course, and start by discussing examples arising from binomial coefficients and partial fractions. They move through topics such as Stirling's formula, Bernoulli numbers, Euler's constant, and the Riemann Zeta function, along with other ideas which arise in trigonometric functions, exponential functions, and logarithmic integration. One of the more interesting chapters discusses the integral of the function e-x2, which the authors call the "normal integral". There is also a nice appendix describing the Wilf-Zeilberger method, which can be used to evaluate finite sums. It is not surprising that evaluating difficult integrals often involves difficult, and interesting, mathematics, and the authors succeed in presenting the material not just as a series of tricks but as an interesting and cohesive area of mathematics.
However, while I found many of the topics in the book to be quite interesting, I found it overly cumbersome to read. The authors have no qualms about using Mathematica extensively in the text of the book, and, while in the introduction they claim that the reliance on it will be minimal, I found that even to a casual user like myself it was often difficult to follow what they were doing. More importantly, the text jumped between exposition and examples and exercises and experiments (of the Mathematica sort) rapidly and in such a way that I often found myself getting confused about what the authors had shown and what they wanted the readers to show and where they were going. The book demands extremely active reading, preferably with a computer at your side. This is not necessarily a bad thing in a book, but a casual reader is not likely to devote the required energy to the topic, at least not without significantly more motivation by the authors. And that is my main complaint with the book — the integrals may be irresistible to the authors, but they did not succeed in conveying that level of excitement to me.
Darren Glass is a VIGRE Assistant Professor at Columbia University.