CHAPTER V.

ON SUCCESSIVE INTEGRATION OF AN EXPLICIT FUNCTION

OF ONE VARIABLE.

149.] In the preceding Chapters, so far as integration with respect to any one variable has been considered, the infinitesimal element which is the subject of integration has been an infinitesimal of the first order: and consequently the integral of that infinitesimal element has been a finite quantity. In the general theory of integration however, no restriction is put on the order of the element; the element may be an infinitesimal of any order; and the effect of integration on it will be the reduction of that order by unity. Thus if an element is of the nth order, the first integral of that element will be an element of the (n-1)th order; and the integral of that element will be of the (n-2)th order, and so on; until ultimately, if the integration-process is carried on so far, the nth integral will be a finite quantity.

Now we will suppose the superior limit in all these successive integrations, to be the same general variable x, the inferior limit being constant although arbitrary in each case. It is evident then that each integration will introduce an additional term, which will be a function of the inferior limit. It is convenient to represent this additional term by a constant; and thus, in the entire process, n additional terms, or n arbitrary constants will be introduced; and the final integral is not considered complete unless it contains them.

x

150.] Suppose F(x) to be a function of a finite and continuous for all values of its subject-variable within the range for which we employ it; and suppose its derived functions to be F(x), F"(x),... F" (x), and to be subject to like conditions: then,

F (x)+Cn-21.2 + Cn-1x+Cni

(4)

(5)

and therefore, abbreviating the notation by writing " da" for

SS...

tion,

dx dx, when the latter series involves n symbols of integra

C1, C2, ... C being n arbitrary constants. This is the general expression for the nth integral of an element of the nth order. The following are particular cases of the preceding general form.

xn-1

...

e* dxn = e* + C1
e* + C1 1 . 2 ... (n − 1) + +C-1x+Cn

151.] Many useful theorems of successive integration may be deduced by means of the Calculus of Operations, of which a concise account is given in Chapter XIX. of Vol. I. In the first Chapter of the present volume the reciprocal relation of the differential calculus and the integral calculus has been explained, and the processes of the latter have been proved to be inverse of those of the former. Also the inverse character of the symbols of the latter calculus relatively to those of the former, has been exhibited in Art. 7, Chapter I. Now, as the operations of differentiation and derivation are subject to the commutative and distributive laws, and as the symbols of the operations are subject to the index law, so will the symbols of the inverse operations be also subject to

the same laws; and the transition from the symbols of one calculus to those of the other will consist in only a change of sign of the are the symbols of differentiation and integration respectively, we have as in Art. 7,

adf

index. Thus, as d and

and as these symbols are accordant with the index law,

(7)

(8)

(9)

(10)

These laws are applied in the following examples; care being of course taken to select those cases which satisfy the required laws.

Ex. 2. By Ex. 6, Art. 54, Vol. I, if m = a tan o,

are explicit functions of x, and making n negative, we have, as in

which is Bernoulli's Series for the determination of an integral; see equation (233), Art. 117.

Equation (11) will apparently not give a finite result for the integral, unless the series in the right-hand member is continued to an infinite number of terms; or unless the derived functions of u should vanish after a certain term. The limit of the sum may however be determined by the following process.

d

Considering to affect u only, and dr to affect v only, and in

dx

the right-hand member of the equation separating symbols of operation from their subjects, we have

remembering that the symbol in the left-hand member refers to the integral of uv, whereas those on the right-hand side refer to either u or v according to the preceding hypothesis. In the symbolical form the limit can easily be expressed by means of the general expression for the limit of Maclaurin's Series.

The preceding theorems are deduced from laws of expansion, and from developments which the integration-symbol is subject to, and are thus far independent of any condition as to their subjects. Certain conditions however are requisite, and are the same as those which have already been explained. That is, if the final results are determinate, the integrals must be definite, and none of the elements must be infinite within the range of integration.

153.] The following cases, in which certain subject elementfunctions are supplied, will be useful in the sequel.

both of these formula will be of great use in the sequel.