Now each of these fractions gives an integral of the following form; viz.: 14(x—a)+Bb in A 0227 30 dx = [ log {(x – a)2 +0}+B tan-1270]. sia = BT. Consequently By a similar process it may be shewn that pom dx T . 2m +1 2m +1 3 sin - + ... 1 2m +1 (19) " As very many instances of the evaluation of a definite integral by means of the corresponding indefinite integral will occur in the sequel, it is unnecessary to add others here; and I may observe, that many of the preceding have been investigated, because they will become hereafter the subject matter for illustration of general theorems. SECTION 2.- The Change of Limits in Definite Integrals. — The Resolution of a Definite Integral into two or more connected by addition or subtraction.-Cauchy's Principal Value. 83.] We proceed now to farther researches into the general theory of definite integrals. That our notion of a definite integral may be clear and precise, it must be borne in mind that the symbol / "°*'(x) dx, as well as its equivalent F(Xn) —P(x.), is only a concise expression for the sum of the series given in Art. 4: so that we have **'(x) dx = P(1.2) – F(x0) = (x,-X.) 8 (X.) + (x,— X7) F'(x) +... ... + (x,-Xn-1) F(Xn-1), (21) (20) wherein X1, X2, X,... Xm_y are the values of x corresponding to the (n-1) points of partition of xn— Wo, each part being infinitesimal, and n being infinite; and wherein F'(x) is finite and continuous for all employed values of its subject-variable. Now whenever a definite integral or its properties are the subjects of inquiry, the definite integral must be considered as the sum of the series given in (21), and its theorems are true because they are true of (21). This is the point of view from which definite integrals have been considered in Art. 8, and from which they are always to be considered. Hence it is evident that if f'(x) does not change sign between X, and Xo, but is always either positive or negative, the definite integral of F'(x) dx taken between these limits is likewise positive or negative. If however F'(x) changes sign, the definite integral of f'(x) dx will be positive or negative, according as the positive or negative part of the series is the greater. 84.] Subject to the condition that f'(x) does not become infinite or discontinuous between the limits of integration, the limits may be altered, and the value of the integral will generally be changed thereby. Thus the integral becomes a function of the limits, and may be treated as such. The case of a continuous variation of the limits, and of the consequent variation of the integral, will be considered in Section 3 of the present Chapter. But it is necessary at once to investigate certain simple cases of a change of limits; and in the first place the effect of a reversal of the limits. By the definition of a definite integral, / F'(x)dx = (xn-1—X») f'(Xn) + (xn-2-Xn-1) P'(x,-1)+... ...+(x1 —X,) F'(x2)+(x, —27) F'(x) (22) =-{(x1 —X,)F'(x + x X) + (x2-x) F'(x1+x2–X;)+... ... + (xn-Xn-1) F'(Xn-1 + X , — Xn-1)}. (23) But as X7-X, X, — #1, ... * , -Xm-1 are infinitesimals; by (6), Art. 18, Vol. I, F'(x + x1 —%) = f'(x0) + (x1 - x0) F"(20), (24) 1 *'(x) dx = -{(xz — X.) F'(x.)+(x2—27) F'(x7) +... ... + (x ,—&»–1) F'(Xn-1) +(x; -2.)? F"(x)+(, —x)F"(x1)+ ... + (x n — Wn-1)47":2n-1)}. (25) Now the last row of terms in the right-band member of this equation involves infinitesimals of a higher order than those in the upper rows; and the number of terms is the same in both : these latter must therefore be neglected in the sum; and we have r’x) dx = -{(x,-X) F'(x)+ (%,—X;) F'(x)) + ... ...+(x, —-Xn-1) F'(Xn-1)} =-/**P'(x) dx ; (26) and thus the effect of a reversal of the limits of a definite integral is the change of sign of the integral. It is however to be observed that the two members of (26) are not absolutely identical; but that they differ by a quantity, a say, which is an infinitesimal of an order which must be neglected; so that we have **F'(x) dx = - ***'(x) dx—a; (27) KO where A=(x,-X)"F"(x,)+(x2-27)2F"(x2)+...+(@n - Xn-1)2F"(xn-1).(28) So many applications of this Theorem will occur in the sequel, that it is unnecessary to insert examples here. 85.] A definite integral of which Xn and X, are the limits is equal to the sum of a series of definite integrals of the same element-function, provided that the extreme limits are the same, and the several intermediate limits are continuously additive. Let / **'(x) dx be the integral under consideration ; and let us suppose x,- Xo, which is the range through which the integration is to be effected, to be divided into n finite parts; and to the several points of partition let X1, 82, ... Wn-1 refer; so that we have the identity, Xn-= (*—X)+(x,-&7)+...+(x6,Xn-1). (29) Also let the finite intervals X,—X, X,—X1, ... X. , -Xn-, be divided each into infinitesimal parts; of which let X7-X, contain a parts, to the points of partition of which let az, az, ... Qa-correspond; let x,-, contain b parts, to the points of partition of which let Which results are in accordance with those demonstrated by another process in Ex. 8 and 9 of Art. 82. Various opinions have been expressed by mathematicians as to the correctness of these values of sin oo and of cos co; they are quantities apparently indeterminate, because as the arc is increased by 27, sin x and cos x pass through all values between + 1 and -1, +1 being the maximum and -1 being the minimum; and because there is no reason, à priori, why any one value between these limits should be taken rather than any other. Moral expectation might lead to the choice of the average value, which is zero; but moral expectation is not mathematical demonstration. The indeterminateness too of the value is also inherent in the preceding series; so that this process of evaluation would hardly be considered rigorous; for zero is not the value of either of the series, unless the number of the terms of the series is of the form 2m in the former series and of the form 4m in the latter. When however the number of terms of a series is infinite, what is the form of that number? Is it par or impar? Is it par par, or par impar? On the answer to these questions must an opinion as to the rigorousness of proof of the preceding process depend. And the answer depends on the view taken of infinity. If infinity, which is the superior limit of the integral, is capable of discontinuous increase by units, the preceding process will probably be considered to be wanting in rigorousness; but if infinity admits of only continuous increase, so that 27 may be considered an infinitesimal increment of an infinite arc, then the preceding process will probably be considered sufficient. On the view taken of infinities and infinitesimals, in our theory of definite integration, the process of evaluation given in Ex. 8 and 9, Art. 82, is apparently free from objection. Indeed for every positive value of a, however small, these equations are arithmetically true; and the results may be shewn to be true by actual summation.* 87.] Again, if & is a value of x not included between Xm and Xos but lying beyond the range of integration, say beyond Xn; then, if y(x) is finite and continuous for all values of x between X, and {, by (31) we have * See De Morgan : Differential and Integral Calculus. Page 571. London. 1842. |