40.] Integration of (2 ax-x2) dx, and of (2 ax + x2)3 dx.

[(2 ax−x2)* dx = [ {a2—(x− a)2}1 d(x− a) ;

f{a2

which latter integral, if x-a = z, becomes {a2—z2}1dz, and is therefore of the form (79); and we have

[(2 ax+x2)1 dx = [{(x+a)2 − a3}1d(x+a)
= f (z2—a2)1 dz, if z = x+a;

=

=

=√

a2

2 log {x+a+ (2 ax+x2)1}. (83)

=

dx

(a + bx + cx2)

Hence

=

(+1)

1

2 a2

=

k2;

1

dx

612

4ac-b2)

+

4c2

{(x + x c

2

dx = dz;

whence we have

(84)

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.

p

To find the integral of xTM (a+bx") dx, where m, n, p, q are constants, integral or fractional, positive or negative.

929-1

+(-a)
(za— a)

nb"

1

m + 1

is an integer, (za—a) is of a rational

1-n

dz;

m+1

1

(z — a)

dz. (86)

nb n

m+1

-1

n

n

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.

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

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

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

b+ax¬" = 29,

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

+

n

is an integer, by substituting a + bx” = za ;

is an integer, by substituting b + ax¬"=z? ̧*

* For other methods of Rationalization, and indeed for a complete collection of integrals of all kinds, the reader is referred to "Sammlung von Integraltafeln," von Ferdinand Minding; Berlin, 1849.

Let a2+x2 = z2; .. x = (z2 — a2)3; dx =

z dz
(z2 — a2)

1, and is integral.

q

= 22;

x2 dx
(a2 + x2)3

=

=

=

3a2 23

203

=

3a2 (a2+x2)

dx =

46.] Integration of = {2, (a+b), (a+b), dr

R

ex

where R is the symbol of a rational function.

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

where R(2) denotes a rational function of z.