Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/46

INFINITESIMAL CALCULUS This result can be also readily obtained by aid of tlic symbolic theorem of 90, thus, F(a>) D- f - the remaining terms being neglected since D"+ 1 F(a:) = 0. This result plainly coincides with that previously found. More generally, if Y(x, c ax , c bx , . . e**) represents a rational integer function of a 1 , e"*, c bx , . . . the integral of Y(x, e ax , c bx , . . . e**}dx can be determined. For, this function, being composed of products of integer positive powers of x, c ax , &c., will consist of a number of terms of the form Ax m c lax c nbx ... or Ax m c( la+ " b + Y, each of which can be integrated by the preceding method. Again the form fF(x, ogx)dx, is reducible to f(c z ,z)c z dz, by making x = c z , and consequently, when F represents an integer algebraic function, is integrable by the method considered above. 117. We next proceed to give a brief account of the treat ment of the integral /l^ dx, in which f(x) and <f>(x) are rational J 4>( a ) algebraic functions of x. This class of integrals early engaged the attention of mathema ticians. For example, Leibnitz and John Bernoulli, in the Acta Enulitorum (1702 and 1703), showed that such integrals depended on the method of partial fractions. The processes there given were simplified and generalized by Euler (Introductio in Analysin litfini- torum, 1748). When the degree of f(x) is not less than that of <f>(x), the ex- f(x) pression t-i ( can by division be reduced to an integer along with <f>(x) a fractional part ; we may, therefore, suppose that we have reduced the degree of /(,*:) to less than that of <p(x). Then, a being a simple root of (p(x) = 0, we may assume <t>(x) = (x-a)x(z), where x( x ) is not divisible by x - a. If we now make (#) A (x - a) x (x) x) + (x-a)f 1 f(x)-Ax(x) we have This gives In order that the second member should be an integer expression f(x) - AX(X) must be divisible by x - a ; hence we get In like manner, if b be a second simple root of <f>(x] 0, and con sequently a root of x( x ) = 0, we may make x(%) = (* - ^)<K^ )- Hence we et x-b from which it follows that B. M Finally if a, b, . . . I represent all the roots of which are equal, we shall have , no two of = _ i_ <t>(x) x - a x - b x-l where Hence In the general case of multiple roots, we may suppose and assume A, A a , . a; 6 (a; 6) 3 The constants A 1; A 2 . . . Bj, B 2 . . . Lj . . .L A , can be determined by-ordinary algebraic methods, and each term is immediately in tegrable. The preceding is called the method of integration by decomposition into partial fractions. The method here given applies also to the case where <(. ) = has imaginary roots. In that case it is usually, however, simpler to employ a somewhat different treatment. Thus, to a pair of imagin ary roots a2/3 corresponds a partial fraction of the form Ls + M Also, for n pairs of equal imaginary roots, we have additional terms of the form L,.y + M.. Lyg + M., L+M,, Each of these expressions consists of two parts, one of which can be immediately integrated. For example, (L, t ,-y+M ra yfa _ L H (x-a)dx (L^+MK/-_ _ the former can be at once found ; the consideration of the latter class of expressions is postponed for the present. Many integrals of the form here considered may be determined by a transformation, without the employment of the method of partial fractions. For example, / J ax- 1 + 2bx n + c is at once reduced to an elementary form by making ce" = ~. /" dx is reduced to depend on by making r = x-b Ex. 1. To find then x 5 dx = j^zdz, and we get x(a + ox") n J az + b tin Ex. 4. To find Here 1 /%v + 2 cos },e)dx 1 /~G >; - 2 cos Q}d 4 cos ^Qj 1 + 2x cos -^0 + x 1 4 cos $J 1 - 2,r cos ^0 + x 1 - 2u;cos Jj Ex. 5. Find the integral of 1 +2.* fos -10 + a; 2 1 , 2-> sin !,0 tan ~ 2 4 sin A0 l-x- a + 2/r + ex- 1 when aol" . It is easily seen from the last that its value may be written 2*A- 4- .v- ~~f"-+ 2 . . 2x /b + K~ = tan /= ;; /-