INFINITESIMAL CALCULUS and, since x may have any value, we have in general tf*) -!"(*). Accordingly we may write Hence the process of integration is reduced to the determination of a function F(x) when its derived function F (a ) is known. We shall illustrate these preliminary remarks by one or two examples. Ex. 1. Find the limit of the sum of the scries f- -7, , . . . + % 2 + l 2 w 2 + 2 2 n 271 2 when n is indefinitely increased. Let dx = , and the limit of the series is easily seen to be re- n presented by dx . TT . d ,, . 1 -5 , or is -j- since -=- (tan - 1 x) - -., 1 + a? 4 dx ^ 1 + x* Ex. 2. Find the limit of the sum H r- when n is indefinitely increased. /-i Here the required limit/ i > 9 w .1 x" 108. We might have started from the preceding result as the definition of the integral calculus, and regarded this calculus as the inverse of the differential. Thus, as in the differential calculus we investigate the rules for proceeding from any primitive function F(x) to its derived function F (a;), so in the integral calculus our object is the converse, viz., to determine F(,r) when F (x) is given ; or, in the language of Newton, "to find the fluent of a given fluxion." It may be here remarked that it has been shown from geo metrical considerations, in 23, that such a function always exists. In the differential calculus rules have been laid down for the method of determining the differential of any function. There are, however, no direct rules for the inverse process, except by retracing the steps by which the derived has been deduced from the original function. Accordingly, the integral calculus is based on the differential, and to each result in the differential calculus corresponds another in the integral. Moreover, as F(x) and F(x) + C (where C is any arbitrary quantity that does not vary with x) have the same differential, it follows that to find the general integral of F (x)dx we must add an arbitrary constant to F(x). 109. The following elementary integrals (omitting arbitrary con stants) are easily arrived at, and are called fundamental integrals, to which all others that admit of integration in finite terms are ultimately reducible excluding higher transcendental func tions : /sin xdx= -cos x , /cosxdx=sin /dx f dx 2 = tan x , I r-s = - cot x cos*x J aui-x dx . , x - = sm dx /"dx 1 , , x /. * /-5-r 9 = tan" 1 - , /a x ax = -. . J a- + x* a a J log a 110. A number of integrals can readily be reduced to one or other of the above forms. A few elementary cases, such as frequently occur in practice, are here given. We commence with the integral (1) f_ !*e
- * / / T7 ~^ *
Here L_ _ = _J_ (_L_ _L_ (*-o)(*-j8) a-&x-a X-&J (2) More generally, the integral r dx a + 2bx + ex- may be written in the form (ex + 6) 2 + ac - b" or, substituting z for ex + b, The form of this integral depends on the sign of ac - IP. If ac -b 2 >0, we have -1 J a + ~2bx + ex* ex + b If ac - b"< 0, the integral comes under (1), and we have (3) Again, since Z + mx m we have (l + mx)dx _ m /" (b + cx)dx Ic - mb f dx i + 2bx + cx 2 cja + 2bx + cx- c ja + 2bx + cx- The integral of is integral has been obtained in (2). (4) Next, to find / ^ J sin x cos x Here c 2 ), and the latter
sin x cos x J tan x cos 2 * In like manner, (T) = lo& (tan tan x < /-.-/- sin - cos -~- Hence we et (6) Again f tan 2 * dx =y sec 2 * dx -fdx = tan x - x. 111. The number of independent fundamental formuhe must ultimately be the same as the number of independent kinds of functions in analysis. The ordinary elementary functions may be briefly classed as follows : (1) algebraic functions, powers and roots, such as x m , for fixed numerical values of m, &c. ; (~2) trigonometrical functions, sin x, tan x, &c., and their inverse func tions, circular functions, sin" 1 *, tan~ J *, &e. ; (3) exponentials a x , &c. , and their inverse functions, logarithms. Several other transcendental functions have been introduced into analysis, such as elliptic and hyper-elliptic functions, gamma- functions, and others. We propose subsequently to give a short account of the elementary properties of some of these functions. 112. The reduction of an integration to one or more of the pre ceding elementary forms is usually effected by one or other of the following methods : (1) transformation to a new variable ; (2) integration by parts ; (3) decomposition into partial fractions ; (4) successive reduction ; (5) rationalization. Examples of these methods will appear in succeeding paragraphs. 113. The method of integration by substitution corresponds to a change of the independent variable. We shall exemplify it by a few simple cases. Ex. 1. Let u = I , m being a positive integer. J (a + bx)" Assume a + bx z, and the integral transforms into If (z - a) n be expanded by the binomial theorem, each term can be separately integrated. r dx Ex. 2 u = / T- , . J (x + a) Va + 2bx + car Let x + a = , and the integral transforms into iv - J Va + 2b z + e z- where a = c, b =*b-ca, c *=a- Iba + Ca 2 . Ex. 3.
