Page:Encyclopædia Britannica, Ninth Edition, v. 13.djvu/51

~A) (a + 2 In like manner, from Ex. 5, 117, we get INFINITESIMAL CALCULUS we substitute X+k-z for ,r t we get /(x+/o-.AX)=/"/ (XH- ./o Integrating by parts, we have /f(X + h - z)dz = z/ (X +h - ,:) + / hence /(X + 70 -AX) = 7</(X) -K/"/"( X + h - z}~dz . Again, 41 It may be noted that if F(.TI ) approaches a finite value F(oo) as ,r approaches oo the derived function F (.r) must vanish at the same time. 1 :>">. As a further example, let us consider the definite integral where <p(x) is an arbitrary polynomial of the degree n in ,r, and X,, is the coefficient of a" in the expansion of (1 - 2 It has been shown ( 56) that x - -J A "~2. 4. 67772^ Again, by the method of 116, we have --- dx .> ) = by hypothesis, and when the limits + 1 and - 1 are substituted each term in vanishes separately ; hence we have From this it is readily seen that so long as m and n are unequal we have 136. There are many integrals which are capable of being deter mined between certain definite limits without any previous know ledge of the corresponding indefinite integral, and even in cases where the consideration of the indefinite integral would lead to the introduction of a higher transcendental function. Examples of this class will be met with further on. 137. Next, reverting to our original definition ( 106), we have J^ /( i a;)^ = lim.[(j 1 - x^f(x^ + (x. 2 - xjffa).. + (X - Xn-i}f(x n -i)], in which f(x) is supposed to be continuous between the limits x a and X. If now A represents the least and B the greatest value of /(;) between these limits, it is plain that is greater than (X - a. ,,)A, and less than (X - a- )B. Hence f*f(x)&K = (X - a^M , -/ o where M > A and < B . Again, when f(x) is a continuous function, in passing from one limit to the other it varies so as always to lie between the values A and B. Consequently for some value f , of x, we must have/(|) = M , where | lies between x n and X, i.e., | is of the form x + 0(X - x ), where 8 is positive and less than unity. Hence, whenever /(a:) is finite and continuous between the limits x and X, we have ~ X (x - *) / { .r u + e(X - *) | 111 like manner it is shovn that f X /M0M <& =/ { * + <K x - a- ) j- / 3 M> ./-^fl provided /(a;) and <f>(x) are finite and continuous between the limits x a and X, and 0(a:) has always the same sign between these limits. 1 f(v} For example, let 4>(x} = , and write/(,r) instead of i/ -^--, then , Xj- - . S o - in which we suppose that X - a, x - a have the same sign, and lies between x and X. In particular, if = 0, we have lo . 138. Taylor s Theorem. The method of definite integrals fur nishes us with a simple demonstration of Taylor s series. For, if in the conation /-X+A f(x}dx f (X + /,. - S )xlz = --- f" ( X ) +j /"(X + h- =) and so on. Hence we get finally Accordingly the remainder, R,,, after n terms, in Taylor s series, is represented by the definite integral This value of R H can be identified with that given in 46, for by 137 we have U /"* 7i R = u / n-l/7 z TT_ K " i / . ~ "* ~ u u, > where U lies between the greatest and least values of /<")(X + 7i-c) between the limits and h for 2. Hence, since any value of z between and k may be represented by (1 - 6)h, where 0> and < 1, we have 139. Thus far the function f(x) under the sign of integration has been supposed to have a finite value for all values of x between the limits of integration. Let the indefinite integral oif(x)dx be denoted by F(#), and sup pose f(x) = 00 when x.a, where a lies between the limits X and x ; then, decomposing the integral into two parts, we have Accordingly, whenever F() has a finite and determinate value, we have This result also holds if /(#) becomes infinite at one of the limits, provided F(x) is finite and determinate at the same time. For example, the expression -== - becomes infinite when x = a, and also when x= ; but (Ex. 6, 113) F(a) = ir, F(j8) = 0, f 3 - dx 140. The complete discussion of the exceptional cases in definite integrals is due to Cauchy. We purpose here to give a brief account of his method. Suppose that the function f(x) becomes infinite for the particular values of x represented by x v x. 2 , . . . X H , lying between the limits of integration ; then we have /*x /*- r i r x -i /~x /"/(HI)**- f l f(x)te+ f l f(.r}d.r ... + / f(x},lx Jr. Jx* A Jxn f Tl fll f(a:)dx+/ Xl ^f(x}dx+ . . f(x)dx = lim. of -x + where e denotes an infinitely small quantity, and /xj, r j; yu 2 , v.,, . v n , are positive, constants, but arbitrary. In addition, if the limits X and r a become + <x> and-oo, write xnr. 6