At first sight the integration of functions seems to depend as much upon luck as upon skill. This is largely because the teacher or author must, in the early stages, select examples which are known to ‘come out’. Nor is it easy to be sure, even with years of experience, that any particular integral is capable of evaluation; for example, xsinx can be integrated easily, whereas sinx/x; cannot be integrated at all in finite terms by means of functions studied hitherto.

The purpose of this chapter is to explain how to set about the processes of integration in an orderly way. This naturally involves the recognition of a number of ‘types’, followed by a set of rules for each of them. But first we make two general remarks.

1. (i) The rules will ensure that an integral of given type must come out; but it is always wise to examine any particular example carefully to make sure that an easier method (such as substitution) cannot be used instead.

2. (ii) It is probably true to say that more integrals remain unsolved through faulty manipulation of algebra and trigonometry than through difficulties inherent in the integration itself. The reader is urged to acquire facility in the normal technique of these subjects. For details a text-book should be consulted.

Polynomials. The first type presents no difficulty.

Appreciation for help received was expressed in the Preface to Volume I, but I would record how much deeper my indebtedness becomes as the work progresses.

I have been fortunate in the help received during the preparation of this work. The manuscript was read with great thoroughness by Dr Sheila M. Edmonds, of Newnham College, Cambridge, whose criticisms and suggestions were of great value and kept me firmly in the paths of rigour. Dr J. W. S. Cassels, of Trinity College, Cambridge, read the proofs and drew attention to a number of slips. To both I would express my sincere thanks.

A number of pupils helped me in the preparation of the answers. Special mention must be made of Mr J. E. Wallington and Mr P. A. Wallington, who, acting almost as a committee, provided me with a complete set of checked answers; any slips that remain must be due to my own carelessness in transcription. I am deeply indebted to them for a very substantial piece of work.

As on former occasions, I have been greatly helped by the staff of the Cambridge University Press, and I should like to place on record how much I owe to their skilled interpretation of the manuscript.

The Examples come from many sources—the Oxford and Cambridge Schools Examination Board, Scholarship examinations in the University of Cambridge, and Degree Examinations in the Universities of Cambridge and London. I am grateful for permission to reproduce them.