b sin x (2n–3)a s dx (n-1)(62 --a?)(a+bcosx)"-1-(n-1)(62 — a>)/(a + bcos x)"–1 n-2 r dx * (n-1)(62a2)J (a +6 cos x)"–2. " By which means the integral becomes ultimately reduced to s dx Ja+b cos x' the value of which has been determined in Art. 67. 79.] Many of the algebraical functions which have been integrated in Chapter II may by substitution be transformed into circular functions, and in some cases the integrals may be determined with greater facility; and by a reverse process many of the circular functions which have been integrated in the present Chapter may be transformed into algebraical functions. The method is exhibited by the following examples : (n-1)(n-3) (n-1)(n-3)...4.2 :- n(n-2)(n-4)...5.3 cos o }; (58) or to 1- cos 0 (sin 0)"-1 0-1 n n(n-2) COS cos 0 (sin 0)n-s (n-1)(n-3) n(n-2)(n-6, cos 0 (sin 6)"-5 –...... (n-1)(n-3)...3.1 n(n-2)(n-4)...4.2 0 ; (59) and replacing o in terms of x, the results are identical with (90) in Art. 49. 80.] This and the preceding Chapters contain an account of almost all the known methods for finding indefinite integrals. Very few indeed they are, and they may be reduced to two or three general heads; so that most of the labour consists in transforming given element-functions into other and equivalent forms of which the integrals are known. Should any one ask why the number of known indefinite integrals is so small, the reply is easy: we have no means of expressing them ; our materials fail : it is not because the Calculus as a system of rules for integrating and disintegrating (or differentiating) fails; but it is because the materials, on which it has to operate, fail. Other functions and other combinations of variables are required beside those which we now have. A word or two will shew how this is. On an examination of the several forms of functions to which differentiation leads, it will be seen that certain forms do not occur. Thus dx (a, +2, % + + ag.x3 +24x4)#dx, — I e* dx, (a? — x2)+(a? – e22) {1-e(sinp)2}#do, are forms of functions, to which differentiation does not lead; that is, we do not in the Differential Calculus meet with the functions of which these are the derived functions. Also again, when differentiation is performed on a given function, in most cases it changes the nature of it, and reduces it from a more complex and transcendental to a more simple form : thus log x is by differentiation changed into (x)-1dx, that is, into an algebraical form; sin-1 x, tan-1 X, .... similarly give rise to algebraical expressions : in the reverse process therefore of integration the simple functions are changed into more complex ones; algebraical functions may not produce other algebraical functions, but may become logarithmic or circular. In order then that logarithmic and circular functions should generally be integrated, there must be other transcendents higher than they are, and of which they are the typical infinitesimal-elements : but such functions do not as yet generally exist; and until they have been discovered, studied, and had their values calculated and tabulated for given values of their variable subjects in the same way as logarithmic and circular functions have been treated, it is vain to seek for indefinite integrals of the (at present) highest transcendents. Whenever therefore in the sequel we meet with the expression “cannot be integrated,” let the exact force of it be borne in mind; it is not meant that the infinitesimal element-function to which the expression is applied is not the element of some finite function, for doubtless such a primary function exists, and it may be a question of time only when functions will have been examined with accuracy sufficient to have their values tabulated and their properties understood : but it is meant that such an infinitesimal function is not the element of any circular, logarithmic, or algebraical function which has already been the subject of complete analysis and examination ; and thus that the integral cannot be expressed in terms of the ordinary functions or symbols with which we are familiar. Many instances of this incomplete state of the science will occur hereafter. CHAPTER IV. ON DEFINITE INTEGRATION AND ON DEFINITE INTEGRALS. SECTION 1.- Definite Integrals determined by means of Indefinite Integration. 81.] In most future cases of the application of the Calculus, and indeed in all problems into which integration enters, the solution depends on one or more definite integrals; for most of our problems depending on a continuous law, which can be expressed only by an element-function, the results of that law can be definitely found only when the definite integral of that element-function can be determined. Hence arises the importance of definite integration, and of the evaluation of definite integrals. In Chapter I. the general notion of a definite integral has been explained, and some general theorems of such integrals have been demonstrated. I propose in the present Chapter to investigate other theorems, and chiefly with the view to the evaluation of the integrals. Great care will be needed in the inquiry, as the subject is of a very delicate nature; for a definite integral is the sum of a series, and the usual analytical difficulties as to convergency and divergency are inherent in it. The terms of the series are also infinitesimal, and the number of them is infinite, and these quantities, various as to their orders, cannot be subjects of combination and of calculation without considerable risk of error. Neither does any general method exist for the evaluation of definite integrals; a method universally applicable is a desideratum : it will appear in the sequel that certain methods are adapted to particular forms of integrals, but no general principle has as yet been discovered which includes all these several processes. The difficulty of the inquiry is hereby increased, because there is no general rule to which all the cases must conform. Í have however had the benefit of consulting the great work on |