Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/49

 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 &amp;lt;/&amp;gt;(,&amp;gt;) is at most of the degree n - I in ,v. For, if we difi erentiate the expression x m fa + &quot;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 , &amp;gt;&quot; (fe fj-h- ^ + mo/ -^- . Hence, making m = 0, 1, 2, 3 . . . in succession, it is easily seen that /&quot;^L^: i s expressible in terms of /^ and of an algebraic j vx y/x expression of the form ^(cc)VX, where &amp;lt;/&amp;gt;(.&amp;lt;) is of the degree &amp;lt;H - 1 at highest. Again, by the method of partial fractions the integral &quot;/(-&amp;gt; ) 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 + &amp;lt;f&amp;gt; (x)(a + 2bx + ex&quot;) + $(x)(b + ex}. Hence, by equating coefficients of like powers of a-, the value of o and of the coefficients in &amp;lt;f&amp;gt;(x) can be determined. For example, let it be proposed to find A Va + 2bx + ex* &quot;Writing A + 2/j.x + i&amp;gt;x- for &amp;lt;j&amp;gt;(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 &amp;lt;f&amp;gt;(x) are rational algebraic integral fc J functions of x. By the method of partial fractions we may write r/,_ _ _ . + 2 &amp;lt;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&quot; 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&quot; 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&quot;, be represented by N, we have Hence, observing that N, -- . . . are independent of .-&amp;gt; diL an* we have &quot; - -? ~ N -- x-a dn x-a dn 2 dxx - a Consequently, the class of integrals here considered depends ulti mately on the integral ~e&quot;*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&amp;gt;(x) &amp;lt;f&amp;gt;(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 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 -, /&quot;cos mx dx ( -&amp;gt;&amp;gt; ft -- I ( y ft -- ^tt/ it j j &amp;gt;jj i&amp;lt;/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 &amp;lt; i  n - I sin mx I I y- I cos -and/ d.r.^- x-a J x-a and, consequently, on the forms /&quot;sin s dz . /&quot;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 &quot;a slight modification, the logarithmic integral can be written in the form
 * J *(*)