6 Methods of integration
6.1 Concepts
To calculate the value of a definite integral it is normally necessary to evaluate the
corresponding indefinite integral; that is, given
we need to find the function F(x)whose derivative is f(x) 1 = 1 F′(x). We saw in
Section 4.6 that every continuous function can be differentiated by the application
of a small number of rules. Each such rule, on inversion, provides in principle a
rule of integration. In particular, the chain rule and the product rule provide, on
inversion, the two principal general methods for calculating integrals; the method
of substitution (Section 6.3) and integration by parts(Section 6.4).
The aim of these general methods of integration, and of the other particular methods
discussed in the following sections, is to reduce a given integral to standard form; that is,
to transform it into an integral whose value is given in a table of standard integrals.
A standard integral is simply an integral whose value is known and whose form can be
used for the evaluation of other ‘non-standard’ integrals. The most elementary standard
integrals are those given in Table 5.1, and a more comprehensive list is given in
the Appendix. Extensive tabulations of standard integrals have been published, and
it is our purpose in this chapter to show how (some of ) these standard integrals
are obtained and, most important, how others can be reduced to standard form. It
must be remembered, however, that there are many functions whose integrals cannot
be expressed in terms of a finite number of elementary functions. In such cases,
approximate values are obtained by numerical methods; with the development of
computing machines, numerical methods of integration (numerical quadratures)
have become routine and accurate, and some of the simpler numerical methods are
discussed in Chapter 20.
We consider first the use of trigonometric relations (Section 3.4) for the integration
of some trigonometric functions.
6.2 The use of trigonometric relations
Table 6.1 contains a number of integrals that can be evaluated by making use of the
trigonometric relations (3.21) to (3.26) discussed in Section 3.4. The useful forms of
these relations are
(6.1)
sin cos (6.2)
21
2
xx=− 12
cos cos
21
2
xx=+ 12