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SECTION 4.-Integration of Irrational Functions by
Rationalization. 43.] Many infinitesimal elements involving irrational quantities may by a judicious substitution be transformed into equivalent integral and rational functions, and consequently integrated by the methods which have been investigated in the first two sections of the present chapter; the process of such transformation is called Rationalization, and we proceed to inquire into the conditions requisite for its application in certain cases.
To find the integral of rm (a + bxcm)ī dx, where m, n, p, q are constants, integral or fractional, positive or negative.
(27-ain Let a + bx;" = %?;
form : and if m
:-1 is positive, (29– a) " may be expanded by the Binomial theorem, and each term of it having been multiplied by zp+q-1 may be integrated by means of Art. 11. And if + -1 is negative, the integration may be accomplished by means of Section 2, and chiefly by the Reductionformula of Arts. 27–30.
44.] Again, as the element-function wm(a + b**)7 dx may be written in the form c***(6+ ax-»)dx ;
(87) and as this is the same as that of equation (86), it follows that by substituting
b+ax-" = 29, the result will be rational if " + + 2 is an integer; and we
na shall be able to integrate by known methods. Hence we may by means of rationalization determine
f1z(a + bamáda, (1) when" is an integer, by substituting a + bx * = zo;
(2) when“ +? is an integer, by substituting b+ ax-*=27.*
ng 45.] Examples of the two preceding articles.
* For other methods of Rationalization, and indeed for a complete colleetion of integrals of all kinds, the reader is referred to “Sammlung von Integraltafeln," von Ferdinand Minding; Berlin, 1849.
a + bx19
c+ex!? Ic+ex! where R is the symbol of a rational function.
Let I be the least common multiple of the denominators of the fractional indices; and let us assume a + bx
cm - a = z; ii. x =
1(bc-ae)zl-1 C + ex
(6-ez'j2 a+bx Platbariz r Also c+ ex) = z?, (
ctenl = 2', ......; all of which are rational; and therefore
1c + ex! ' c+ ex where R,(z) denotes a rational function of z.