(15.) Development of ƒ xm-1(a+bx")" by series.

-1

Sxxm−1(a + bx1)2 =

b

=

p(p-1)x2n

b3

+

m + 2n

3n

P(p-1)(p-2) x3n

1.2.3.a3

+ &c.}

m + 3n

in which e, e1, &c. are functions of a, a, &c. b, and c.

Let the values of b and c be determined from the equations

of R1(9) may

If R, is an integral function, each term of

1

Y

be integrated, or reduced by preceding methods to the form

If R1(y) is a fraction, it may be decomposed, and each separate fraction may be integrated, or reduced to the form

by multiplying the numerator and denominator of the fraction by the denominator.

R(x)

(a + a1x2 +a2x1+a ̧∞)‡ may be separated into two parts

R(x)

X

by separating R(x) into R,(x2), and a. R2(x2), and

x. R2(x2)
J. (a + a1
(a + a1x2 + α ̧x1)¥

to reduce the first of these, assume

(a + a1x2 + a2 x1) = xy;

then by substituting for a its value, we obtain respectively

204

ƒ2R(x). a+a ̧x2 +a ̧11 and ƒ, R(x). a + a1x2 + a2æ3 | ±2 may be transformed in the same manner as the preceding

formula.

f, |

To reduce ƒ ̧ R(x). a +a ̧∞ +a ̧æ2 +a ̧æ3±3 assume

a+a ̧x+α ̧x2+α ̧æ3]=a*+xz, or=a ̧3x+x,
7

by either of which assumptions, the term a,3 will disappear from the radical.

R(x)

, separate

R(x) into R, (2) and x.R2(x2), and put y=x2: the part corresponding to R() may be further reduced by assuming

R(x)

(17.) Integration of (a+a ̧x2 +a ̧æ1)}

Let a + a ̧x2+α2x2 = (b1 + c1x2) (b2+€2x2),

2

=b1.b2(1+e ̧2.x2)(1 + e22.x2).

2

[1] Let the quantities e,, e, be very unequal, and e1 > e; assume e1 2x2=y2, then

and since (1+2y) may be expanded by the binomial theorem

in a series which converges rapidly, if

is small, the required

(1 − ay2).1+ y2

nearly equal, put the given function

2

1

p2 -k, and

1

2

R(x)

)(2

R(x)

+k, then by substituting

these values we obtainƒ, { (r2 + x2)2 — ko } }

a series which converges rapidly if 2 is positive, and ✯ small: the required integral is thus made to depend on

where p2y2 and q2y2 are either both positive or both negative, and

+ or

2

according as c, and b, have the same or different signs: and the second value of q must be taken, if c1b, and b,c, have different signs.

2

2

&c.

2

where Qm represents {(1+p.y) (1±q%·ym)}* :

or the values of P1,91, &c. may be thus calculated; let

then

P19919

by this process we obtain a series of the form

SR(y2) + ƒ ̧ ̧R(y,3) + &c.

R1(y)

in which, since pq continually increases, P~q, may be made as large as we please.