40.] Integration of (2 ax-x2) dx, and of (2 ax + x2)1 dx.

[(2 ax−x2)1 dx = [ {aa—(x − a)3}+d(x−a) ;

which latter integral, if a—a = z, becomes ({a2 — 22}1dz, and is

therefore of the form (79); and we have

(2

f{2ax+x2)1 dx = {{(x+a)3 — a2} *d(x+a)
= f (z2 —a2)1 dz, if z = x+a;

the latter term of which has been integrated in Art. 37.

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 xTM (a+bx")" dæ, where m, n, p, q are constants, integral or fractional, positive or negative.

panded 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

m+1

n

-1 is negative, the integration may be accomplished by means of Section 2, and chiefly by the Reductionformulæ of Arts. 27-30.

P

44.] Again, as the element-function (a+bx") dx may be written in the form

m+

пр

p

X a (b+ax-")dx;

(87)

and as this is the same as that of equation (86), it follows that by

shall be able to integrate by known methods.
Hence we may by means of rationalization determine

n

SxTM (a + bxTM‚¶dx,

is an integer, by substituting a+bx" = 2;

(1) when

m+1

*For other methods of Rationalization, and indeed for a complete collec

tion of integrals of all kinds, the reader is referred to "Sammlung von Inte. graltafeln," von Ferdinand Minding; Berlin, 1849.

bx

46.] Integration of ■ {~, (a+br), (a+b)? { dr.

R xu,

c+ ex

where R is the symbol of a rational function.

c+ex

} dx,

Let be the least common multiple of the denominators of the fractional indices; and let us assume