The idea of writing a book on all the areas of mathematics that appear in the evaluation of integrals occurred to us when we found many beautiful results scattered throughout the literature.

The original idea was naive: inspired by the paper “Integrals: An Introduction to Analytic Number Theory” by Lian Vardi (1988) we decided to write a text in which we would prove every formula in Table of Integrals, Series, and Products by I. S. Gradshteyn and I. M. Rhyzik (1994) and its precursor by Bierens de Haan (1867). It took a short time to realize that this task was monumental.

In order to keep the book to a reasonable page limit, we have decided to keep the material at a level accesible to a junior/senior undergraduate student. We assume that the reader has a good knowledge of one-variable calculus and that he/she has had a class in which there has been some exposure to a rigorous proof. At Tulane University this is done in Discrete Mathematics, where the method of mathematical induction and the ideas behind recurrences are discussed in some detail, and in Real Analysis, where the student is exposed to the basic material of calculus, now with rigorous proofs. It is our experience that most students majoring in mathematics will have a class in linear algebra, but not all (we fear, few) study complex analysis. Therefore we have kept the use of these subjects to a minimum. In particular we have made an effort not to use complex analysis.

The goal of the book is to present to the reader the many facets involved in the evaluation of definite integrals.