226 
DERIVATIVES AND INTEGRALS 
[VI 
115. Algebraical Functions. We naturally pass on next to 
the question of the integration of algebraical functions. We shall 
confine our attention to explicit algebraical functions (Ch. II, § 16). 
We have to consider the problem of integrating y, where 
y is an explicit algebraical function of x. It is however con 
venient to consider an apparently more general integral, viz. 
j R (x, y) dx, 
where R(x, y) is any rational function of x and y. The greater 
generality of this form is only apparent, since (Ex. xv. 6) the 
function R{x, y) is itself an algebraical function of x. The choice 
of this form is in fact dictated simply by motives of convenience; 
such a function as 
{x + \j(ax 2 + 2bx + c)]/{x — a/ (ax- + 2 bx + c)} 
is far more conveniently regarded as a rational function of x and 
the simple algebraical function *J(ax 2 -f 2bx + c), than directly as 
itself an algebraical function of x. 
116. Integration by substitution and rationalisation. 
It follows from equation (4) of § 114 that if j"yfr(x)dx = (f>(x), then 
|V 1/(0} f'(t)dt = <f> {/(0} (i). 
This equation supplies us with a method for determining the 
integral of yfr(x) in a large number of cases in which the form of 
the integral is not directly obvious. It may be stated as a rule as 
follows: put x=f(t), where f(t) is any function of a new variable 
t which it may be convenient to choose ; multiply by f (t) and 
determine (if possible) the integral of [f(t)j f'(t); express the 
result in terms of x. It will often be found that the function of t 
to which we are led by the application of this rule is one whose 
integral can easily be calculated. This is always so, for example, 
if it is a rational function, and it is very often possible to choose 
the relation between x and t so that this shall be the case. Thus 
the integral of R(\Jx), where R denotes a rational function, is 
reduced by the substitution x = t- to the integral of 2tR (t 2 ), 
i.e. to the integral of a rational function of t. This method of 
integration is called integration by rationalisation, and is of 
extremely wide application.