Examples 2, 3, 4, 7, 8, of Praxis 48 may be integrated

by this method.

51. The following examples are to be integrated by particular artifices of calculation.

Multiply the numerator and the denominator by a2 + x2,

52. Integration by parts, or the method of Continuation.

Since d.xy = ydx + xdy, therefore Sydx = xy-fxdy.

In the same manner, since d.

=

y

dx xdy

y

therefore

These formulæ, particularly the first, frequently enable us to reduce the fluxion by successive integrations to a simpler form; hence this method is sometimes called the method of Continuation or of Reduction.

A more general formula of reduction is xvdx = x/ydx -fdxfxdx, where x and y are any functions whatever of r.

53. In addition to the elementary forms which have been already investigated, Articles 17 and 18, we shall assume those which will be demonstrated in Ch. 9; viz. that

=

(1.) fdx2ax-x2 = cir. area, rad. = a, v.s. = x.

(2.) fdx √ a2 = cir. area, rad. = a, distance from

x2

centre x.

=a and abscissa = x.

(3.) fdx √2ax + x = rectangular hyperbolick area, axis

(4.) Sdx √x2 — a2

54. Ex. 1. du =

a2 = the same area, if the abscissa is reckoned from the centre.

x2dx

√x2± a2

Qu = x √/ x2 + a2 — a2A, and taking a2 either positive or nega

which is a circular area form (Ch. 9): or it may be reduced to a circular arc form; for

is a positive integer may be integrated by continued re

duction.

Multiply and divide by √x2+a2, and we have