at binti we shall have by division (4). Shew that the integration of depends upon that of ✓1VI; and that off upon that off V1. (5). Reduce the integration of l to that offr V1 in and the integration of S to that of S2o Vi ht? ON THE INTEGRATION OF RATIONAL FRACTIONS. PE 63. Let be a rational fraction (Pand Q being functions of x); Q and let the greatest exponent of « in P be less, at least by unity than in Q. If it be not so, divide the numerator by the denominator, till this latter condition takes place. For exaniple, if the fraction 6 were of which the b Pr second part is such as we have supposed for Q 64. This done, find the factors of Q, as if we had to resolve the equation Q=0; and if they are all of the first degree, real, and unequal, the proposed fraction will then be of the form ar*-*+60*2 + &c.... ta: (x-1) (x-g) (x-h) &c. To find the fluent in this case, we must decompose the proposed Ax Br C: fraction into these others. it + + &c.; of which the in x-g x—h tegral (art. 56) is Al (x-f) + Bl (x-8) +&c. (where l stands for the hyperbolic logarithm of the quantity within the parenthesis.) To this value we must add a constant C, and determine the com efficients A, B, C, &c. by first reducing them to the same denominator, and then transposing, and successively equating to zero the coefficient of each power of x, which process will give as many equam tions as there are unknown quantities. Ex. Required the fluent of x (aa—xx)' x (a +x)(a a IC xic atbp3 or + aa aa aa Therefore A== Ai Bi I decompose this fraction into the following ones: + ata and reducing to the same denominator, transposing, and ordering the terms, I find Ado+Bax+Bux -C *(aa-x) 1 1 1 + of which the integral is 2aa 2aai satu k_1/a_)_!(a+1) + Ic Where le re2aa 2aa presents the correction C, the form of which is optional. 65. This method will always succeed when the factors of the denominator of the proposed fraction are all real and unequal; but it several among them were equal; if, for example, (r--a)" represented any number m of these factors, we must then decompose the fraction into the following ones. A Bi Ax-1+B'-+ &C.... +R +&C.... + (a) and, after having determined the coefficients as before, we may pro Bm ceed to integrate X + * + &c. or in general (x-a)" ritma) më (oma), by making x_a=%. Example. Let it be required to find the integral of Coco ++2) Ai (Bx+C): (Dx+E); + + *(1-1) (2+1)* (1-1) (x+1)* whence A=2, B= D: therefore (x3 + x2+2) (743x) (5x+7)Ë Now to inte *(x-1) (*+1)* (1-1)? (5+1) grate the fraction (7—31) 5. make r-l=%, which changes it into + Ar-1 + 2 and treating in the same manner the other fraction, we obtain for the 3 (x r+1 66. If in the denominator of the proposed fraction P* there Q should be imaginary factors ; on representing one of them by x+a+bv --1, there will be another of the form rta a-bn-1, and their product x? + 2ax + a + b* will be a real factor of Q. We must therefore find the coefficients 2a, a', b, of the factor of the second degree, which is always possible; and the real factor of the second degree 2+ + 2ax + a + b, or more concisely ? +mz+n, will be determined. Then we must suppose that (Ar+B): is one of the partial factors of the proposed fraction, and + m2 +12 determine A and B, as before. After which, making r+am=, the fraction will become A' 2+B) 2 zz +66% A'zz B'2 A'zz A' 1(xe +66'), and S. zz +6b 2 B' B B b Be arc tang +C; and consequently we shall by this means arrive at the fluent required. Examples. ? Az (Bz+C)ż + ; we shall find that A= (1+z) (1 + x2) 1+2 C--, this changes the proposed fraction into these 3 1 3 of which the integral is 2 1+2 12 1+zz 2 (1+z) – 2 (1+z2)+ arc tan z+C. : be proposed, which is reducible to X (1+x) '(1 +3+x1) B 1 2 1+ 2z 2 Again let : _(2x+3) + + To integrate this last quantity, make (1+x)? 'i+x+x.c (34) 2 *=2–, and it becomes of which the zz + zz + 1 2z integral is : 1 (zz+1) arc lan zz + Consequently by substituting the value of z, we have for the com 1 1 plete integral lx—21 (1+x)+31(1+x+xx)+ + arc tan 1+30 73 (25+1) +C. 13 67. .The last case which remains to be considered, is that in which the denominator Q has one or more factors of this form (xx+ax +b)". Here we must suppose that the partial fraction proceeding from this «2m-' + Bxem-? +&c.... +R); factor is (Ax?m-1 t, and then determine the (xx+ax+b)" coefficients as before. After which, making x=2-4a, and substituting, the fraction will become of the form A'zzm-' + Bz2m-*+&c. +R' : ż, which may be thus decomposed 2+&c. (zz+b'b') (z2 +6°6) Now those terms in which the numerator contains any odd power of s, are integrable, in part algebraically, and in part by logarithms (57); and those in the numerator of which z is raised to any even power, being of the form. Mz2k z may be reduced to the form. (zz +6%)m? art (63); that is, we are able to integrate them, partly algebraically, and partly by circular arcs ; consequently we shall by combining these means obtain the integral of the proposed fraction. 68. The following example will serve to elucidate these different methods, as it contains them all. + Ai (Bx+C) : _ (Dx+E)* _ (Fx' +Gx’ +Hr+1).; + + 1+ xx +2 (x2+1) We shall find by reducing these fractions to the same denominator, 1 D F: G: Now to integrate the partial fractions - 1. and (+8+1)* $ we must have recourse to the method of reduction before (ax +1)" explained. Thus by formula (A), article 61, we find / (1+r)* |