II. To every factor of the form (aa)" corresponds a series of partial fractions of the form Any one of the coefficients as M, is given by the equation every factor of the form a2 + ax + b corre IV. To every factor of the form (x2+ax + b)" corresponds a series of fractions of the form To determine M and N let V = Q(x2+ ax + b)"; then the equations A = 0, B = 0 are conditions for finding M and N. If now we put U − (Mx + N) Q = U1, x2 + a x + h where U 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) 2 (r− 1) {(x − a)2 + B2 }r-1 da + (Ma + N) √ {(x − a)2 + ß3}' * The expression for the last integral will be found in the following Chapter on formulæ of reduction. (1) Let U 2x+3 x2 + x2 − 2 x In this case the factors of V are x, x 1, x + 2, and as d V = 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 to 0. 2x2 + 7x2 + 6x + 2 x2 + 3x2 + 2x2 The roots of the denominator are 2, 1 and two equal U V = x2 dx x3 + 5 x2 + 8 x + 4 + log (x + 1). (1) a being even; U (8) Let V a" (x 1 1 + &c. -x)' 72 n N ... 2 (n - 1) 1.2 (n − 1) ... log (1) Murphy, Camb. Transactions, Vol. vi. |