INFINITESIMAL CALCULUS 39 Thus it can be shown that, if Y(x) is an integer rational function nf the degree n, then y Va + in which a is a constant, and </>(,>) is at most of the degree n - I in ,v. For, if we difi erentiate the expression x m fa + "2bx + cx- with respect to ,r, we readily obtain, after the integration of both sides, and the substitution of X for a + Zbx + c, 2 , >" (fe fj-h- ^ + mo/ -^- . Hence, making m = 0, 1, 2, 3 . . . in succession, it is easily seen that /"^L^: i s expressible in terms of /^ and of an algebraic j vx y/x expression of the form ^(cc)VX, where </>(.<) is of the degree <H - 1 at highest. Again, by the method of partial fractions the integral "/(-> ) dx A reduces to terms such as the preceding, along with terms of the form r dx If in this latter we substitute for x - a, it reduces to the form 3 + CZ 2 in which A = c, B = - b - ca, C = a + 2 Ja + ca 2 . 127. Integrals of the form here discussed may also be treated by the method of indeterminate coefficients. Thus, writing X for n + 2bx + cx*, and differentiating the equation at the commencement of 126, we get or F(x) = o + <f> (x)(a + 2bx + ex") + $(x)(b + ex}. Hence, by equating coefficients of like powers of a-, the value of o and of the coefficients in <f>(x) can be determined. For example, let it be proposed to find A Va + 2bx + ex* "Writing A + 2/j.x + i>x- for <j>(x), we get x* = a + 2(a + 2bx + cx-)(/j. + vx] + (A + 2/iX + vx-)(b + ex), from which we deduce 128. Again, if F denote a rational function, the integral jF(x, /ax + b, /a x + b )dx is reducible to the preceding type, by making V this gives = y. For . /, ;/ Vax+b= L/^-- a / and the proposed becomes of the form ff(y, VYXy, in which Y is of the second degree in y. 129. Having given a sketch of the various methods of reduction of integrals to the forms usually regarded as elementary, we proceed to introduce further transcendental integrals by considering the gnz/W^jj } j n w hich f(x) and <f>(x) are rational algebraic integral fc J functions of x. By the method of partial fractions we may write r/,_ _ _ . + 2 <p(x) x-a (x - a) 2 or, making a slight change in the constants, where D stands for the symbol The method of integrating ~F(x)e" x dx has been already considered ( 116). The integral of the remainder depends on that of the ex pression If the symbolic expression A + A^ + A^D 2 . . . +A n D" be repre sented by /(D), this integral, in symbolic notation, is represented by Again if /(-?(), or A- A 1 w + A 2 2 - . . . A ,,7i", be represented by N, we have Hence, observing that N, -- . . . are independent of .-> diL an* we have " - -? ~ N -- x-a dn x-a dn 2 dxx - a Consequently, the class of integrals here considered depends ulti mately on the integral ~e"*dx /- J If we make x- = log z, this integral reduces to the form ( 115, Ex. 4) dz r^L It is impossible to represent this latter integral, in a finite form, in terms of :. it is accordingly regarded as a function mi generis, and is usually styled the logarithmic integral, and sometimes Sold- ner s integral. Its expression in the form of a series will be de duced in a subsequent section. 130. Next, if we replace n by in, where i stands for V - 1, e nx 7W becomes (cos nx + i sin nx) -&4- , 4>(x) <f>(x) and by an analogous treatment it can be proved that integrals of the forms /;os nx A-4- dx all( i / sin nx A^i &x
- () J *(*)
depend on the forms Finally, denoting by F(sin x, cos x) an integer polynomial in sin x and cos x, it can be shown that the integral /ZLil. F(sin x, cos x)dx Q(X) can be reduced to the same fundamental forms. For the poly nomial F(sin x, cos x) can be transformed into a linear function of f(x) sines and cosines of multiples of x. Again, decomposing ^--j-~- by the method of partial fractions, the integral in question can be made to depend on integrals of the form /sin mx dx -, /"cos mx dx ( ->> ft -- I ( y ft -- ^tt/ it j j >jj i</j and consequently on /dx sin mx ( ) - N and /dx cos mx ( ) 7 - . dxj (x-a) J dxj (j--a) These integrals, by the method of 116, depend on r ( d
. r f < i n
- I sin mx I I y- I cos -and/ d.r.^- x-a J x-a and, consequently, on the forms /"sin s dz . /"cos z dz 131. These latter integrals also are now regarded as primary functions in analysis, and are incapable of representation in terms of z except by infinite series. These functions have been largely treated of by mathematicians, more especially by Schlbmilch (Crelle, vol. xxxiii.), by whom they were styled the sine-integral and the cosine-integral. Also, intro ducing "a slight modification, the logarithmic integral can be written in the form
