OF THE INVERSE METHOD OF

FLUXIONS.

In what has hitherto been delivered of the theory of Fluxions, we have supposed the relation of the variable quantities or fluents to be known, and have investigated that of their fuxions. But in the inverse method of fluxions, we propose to ascertain the relation of the fluents from that of their fluxions.

52. To indicate a fluent, or, as it is also frequently termed, an integral, we shall employ the leters; for example, fax, is the general expression of all the quantities which, when thrown into fluxions, produce ai ; and as ac may equally by derived from a x simply, or or from ax +a constant quantity, to each integral we add a constant term C, which must be determined afterwards by the conditions of the problem.

The quantity ax being in a measure the sum of all its elements at, we call the expression s ar, the sum of ar; and to integrate, to sum, or to find the fluent, are synonimous expressions.

If every fluxional expression were derived immediately from some algebraic function, every fluxion would have its integral or fluent ; but as we call any quantity affected with , y, &c. a fluxion, there are many which are not susceptible of integration, because they cannot have been produced by taking the fluxion of any quantity: yi, for example, is of this number.

Many others we have not hitherto been able to integrate but by approximation. Such are the fuxions of logarithms, of circular arcs, and in general of all those quantities which are called transcendental. We shall first treat of those which admit of exact, or algebraic integrals.

OF QUANTITIES SUSCEPTIBLE OF AN

EXACT INTEGRATION

53. Since the fluxion of wis nx"-14, it is evident that the fuent of neonat e must reciprocally be z"; therefore 5 ****and making

S

;

.

gives for the integration of a fluxional expression of one rerm only, a rule the reverse of that given for finding the fluxion of any algebraic quantity consisting of a single term.

54. Therefore to integrate a flucional expression of one term, increase by unity the exponent of the variable quantity, and then divide by the exponent thus increased, and by the fluxion of the variable. 55. This rule, however, is subject to an exception in the case

1 of m=-1; for then the fluent becomes ++C,

+C, that is, it assumes an infinite form. But the Auxion zmi is in this case reduced to which expression we have shewn in article 16 to be the fluxion of the hyperbolic logarithm of x, hence its integral is lz. Therefore Stlx +C, and consequently all fluxional expressions consisting of one term, and containing only one variable quantity, may either be integrated exactly, or at least approximately, by means of logarithms. Several other fluxions may be integrated in the same manner.

56. Suppose y==(a+bx+cx2 +&c.)=x+bxč+cx ?; + &c, and we shall have y=C+as+

br2 cr3
+ar + +&c.

2 3
Let y=ač (6+x)"; if we niake 6+x=%, we shall have <=2, and

m+1 j=az" 2; consequently y=

m+1
(6+2)*+1+c.

+C. When m + 1 m+1

ax m=-1, we shall have y= =a log (6 +x) + C-log c (6+2) by making C=log c. 57. Let now j = armii (6+")", we shall have y=C+

n (m+1) (6+2")*+1; and in general, if we have y=x": (a+bz“), we shall find, on developing this expression, j=a* x* * + kak-b**** it k (k-1)

ak-? 62.22m+" ; + &c. of which the integral is y=C+ 2 k

k (k-1)_at_2 72 m2m +2+1 + 1+1 m+n+1

2 (2m+n+1) k (k-1) (K-2) + ak3 633m+1+1

+ &c. 2. 3 (3m +n+1).

This integral will always be finite, when k is a positive whole number. But we must observe, that if after having expanded the binomial, there occur terms of the form“, we must integrate them by logarithms. The method is the same for *** (a+bat +ex? + &c.)

az

a

bta

ak zh+1

ak-ibm+n+1

58. Hence we may conclude that any binomial fluxion represented by the formula 1"* (a+br) is integrable algebraically.

i I. When n=m-1, whatever may be the values of m and k. II. Whenever k is a positive whole number for all values of m and n. We shall add two other cases in which an exact integration is possible.

n-1 59. Let a +bx" 2 ; we shall have x +1-(2-a)

and taking n+1 6

m

m

n+1

1

xz (z–a)

m

m

the fluxions on both sides xa x=

a fluxion n+1

mb admitting of integration whenever

n+1

is a positive whole number. For example, let the proposed fluxion be r s (a’ +%), which

n+1 gives 2. The transformed equation then becomes S1 z 2 = 3

a (*.

(a*+r")" (4ro-3a"). Consequently Size : (a +x")* = C+ (a*+297" (432–3a*).

s ro * + 60. x* : (a + bx*)4 = xmk tn 3 (6+ar-my, as will appear by dividing the terms within the parenthesis by me, and multiplying the other factor by the same quantity ; but this fluxion is integrable from what we have just shewn, when

mk +n +1

ork—1"+2) is a positive whole number.

3 56

mk .7

m

m

-= 2, and mkt*

For example, let the fluxion be_

or as it may also be

x2 (a + x3) written, ra; we shall have mk+n+1 -(?->

m :(6+as-myt will be :(1+axNow suppose 1 tax

=%, and we shall have for the transformed

1 expression xz*:(2-1); or :-+ -22, the

1

1

a

-1 ; (a+bx)*

When a binomial fluxion has not the conditions which we have enumerated, we must endeavour to reduce it to some other known fluxion ; such, for example, as that of the quadrature of the circle, or of logarithms, &c

If this reduction is possible, it may be effected in the following manner.

A method of reducing the integration of binomial fluxions to that of other known flucions.

61. This reduction will be effected upon the following principle. We have seen (art. 10) that flux uv = uv+vu, and conversely uv=suv+svi, or by transposition, juv=uv-svi. Therefore if we decompose the binomial x* x (a+b.x" } into two factors, one of which is integrable, we shall cause the integration of the proposed binomial to depend upon that of the other factor, which in many cases is more simple than the given Auxion. This mode, which is at once extensive and curious, is called integration by parts.

Among the different ways of resolving the proposed binomial into factors, we shall choose that which diminishes the index of without the parenthesis. This is effected by writing the proposed fluxion thus, x -m+1 27-1 "

By this means the factor 13(a+bx+)* is integrable, whatever be the value of k (art. 57). Now if we consider the factor 2

Hm+1 =-u, we shall have wm-1; (a+bxm)=o and (a+bxm)*+1 therefore, since suv=uv-svi, we shall have x" : (a+br")"= imt1 (a+b.x) nm+1

ficam 2(a+box";4+1 (k+1) mb (k+1) mb but fx-pa+ba® jk+1=Six-(a + bcm jk (a+b2") =a fitam j (a+bx* )* +"(a+b.xm)", therefore substituting this last value in the preceding equation, and collecting into one the terms involving the integral firmi (a + bx")", we find

n-m+1
(1 +

-) St*(a + bxm)* =
(k+1) m
m+1

k+1
(a+b.x") -(n--m+1). Siram : (a + bx^)*

mb (k+1) from which we get (A)

Sæmis (a+b.umje 1+1

k+1 (a + b.ro

-A(n-m+1) fx"-m : (a +bromyle b (km+n+1)

(k+1) mb =v, and

k+1