The functions which we have already met may be summarized briefly. They are the powers of x and combinations of them such as polynomials and rational functions, the trigonometric functions, the logarithmic and exponential functions, and the hyperbolic functions.

The list is substantial, but it forms only a beginning, and many other functions remain for our attention. The work of this section is directed towards the setting-up of certain basic techniques which enable necessary extensions to be made. The question that we pose is less ‘What particular functions are there at my disposal?’ than ‘How can I set about to find such functions when I need them?’ The results are all well established, and it is hoped that the presentation will enable the student to see some ways of extending his mathematical vocabulary while at the same time absorbing standard information.

The plan of these four volumes has been to present Calculus in the spirit of Analysis, but without detailed examination of the properties which belong essentially to the latter. It is natural that analytical ideas should become increasingly pressing as we approach the later stages, and it seems wise to insert a section now on convergence and similar topics lest the processes which form our main theme later should be treated on a purely mechanical basis. As in the book as a whole, however, so here also we shall try to clarify the guiding principles rather than to establish the wealth of detail which the serious student of Analysis must always require.

The aim of these volumes is that they shall together form a complete course in Calculus from its beginnings up to the point where it joins with the subject usually known as analysis. The whole conception is based on considerable dissatisfaction with much that seems rough-and-ready in the basic ideas with which pupils reach the universities, so that almost anything seems acceptable for ‘proof’ which is superficially plausible. Of course the early work cannot be treated with the rigour appropriate to more mature judgement; but I have tried here, however unsuccessfully, to present the subject in such a way that the more exact treatment, when it comes, can follow by natural development, without being forced to return to a fresh beginning which is often felt to be both unnecessary and even pointless. (How many students lose the thread of analysis just because they do not see any reason for the first few lectures and therefore do not give them serious attention?)

The first volume deals with the basic ideas of differentiation and integration. Graphical methods are used freely, but, it is hoped, in such a way that the essential logical development is never far away. The examples at this stage are mainly very simple, and beginners should have no difficulty in acquiring a fluent technique. Integration appears from the start as area and summation, the method of calculation by inverse differentiation being deduced.

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.