II. To every factor of the form (a - a)" corresponds a series of partial fractions of the form

Any one of the coefficients as M, is given by the equation

III. To every factor of the form x2 + ax + b corre

and from the conditions A = 0, B = 0, M and N are found.

IV.

To every factor of the form (x2 + ax + b)" corresponds a series of fractions of the form

the equations A = 0, B0 are conditions for finding M and

N. If now we put

U-(Mx + N) Q

=

U1,

x2 + ax + b

1

where U1 is necessarily an integral function, we can, from the equation

determine M, and N, as before, and so in succession for all the other partial fractions.

The fraction having been thus, by one or other of these methods, decomposed into a sum of simpler fractions, each of them may be integrated separately by known processes, and so the whole integral is found.

dx (Mx+N)

=

M

1

√ {(x − a)2 + ß'}' = 2 (r = 1) { (x − a)2+ ß°}r-1

dx

+ (Ma + N) √ {(x − a)2 + ß3}

The expression for the last integral will be found in the following Chapter on formulæ of reduction.

3x2 + 2x

2,

Here the denominator contains two equal factors (x − 1)2, and the partial fractions arising from these equal roots are

and the fraction corresponding to the other factor (x + 1) is

The factors of V are x + 1, x 1, and 2+2.

Hence

30

-1

22

1⁄2 log {(x + 1) (x2 + 1)3} − } tan−1 x.

(14) Let =

V x2 + x1 + 2x3 + 2x2 + x + 1

Here there are in the denominator two equal quadratic factors (+1); the fractions arising from them are