INTEGRAL CALCULUS.

CHAPTER I.

THE object of the Integral Calculus is to discover the primitive function from which a given differential co-efficient has been derived.

This primitive function is called the integral of the proposed differential coefficient, and is obtained by the application of the different principles established in finding differential co-efficients and by various transformations. In order to avoid the embarrassment which would arise from the perpetual changes of the independent variable, which it would be necessary to effect if we restricted ourselves to the use of differential co-efficients alone, we shall generally employ differentials according to the infinitesimal method explained in the preceding chapter.

When we wish to indicate that we are to take the integral of a function we prefix the symbol. Thus, if

y = ax1

We know that dy = 4ax3 dx

If then, the quantity 4ɑx3 dæ be given in the course of any calculation, and we are desirous to indicate that the primitive function from which it has been derived is ax1, we express this by writing

S sax3 dx = ax1

When constant quantities are combined with variable quantities by the signs + or — we know that they disappear in taking the differential co-efficients, and therefore they must be restored in taking the integral.

Hence in taking the integral of any function it is proper always to add? constant quantity, which is usually represented by the symbol C. Thus, if be required to find the integral of a quantity such as

dy=3ax dr

y = ƒ 3ax* dx

= ax3 + C

where C may be either positive, negative, or 0. We cannot deterize the value of C in an abstract example, but when particular problems are submitted to our investigation, they usually contain conditions by which the value of C can be ascertained. This will be clearly seen when we treat of the applications of the integral calculus.

By reversing the principal rules established for finding the differential coefficients, or differentials of functions, we shall obtain an equal number of rules for ascending to the integrals from the derived functions. Recurring therefore to these we shall perceive that

I. The integral of the sum of any number of functions is equal to the sum of the integrals of the individual terms, each term retaining the sign of its co-efficient. Thus, it

The integral of a function raised to any power is ouined by adding unity to the exponent of the function, and dividing the function by the exponent so increased, and by the differential of the function

for these can all be reduced to the form az dz. Thus,

This formula is very extensive in its application, since we have all integrals of the form ((x))" d.¢(x) composed of two factors, where the one is the differential of the other which is within the bracket.

Ex. 12.

Let

dy = (x2 + y2)* (6x* dx + 4y 1y)

2+ y = 2. 3.x dx + 2y dy dz.. 6x dx + 4y dy = 2dr

III. The above rule fails when =— - 1, since in that case we should find

Sz-1 dz = α, but this arises from the circumstance that the integral belongs

The integral of every fraction whose numerator is the differential of its denominator, is the logarithm of the denominator.

IV. The integral of every fraction whose denominator is a radical of the second degree, and whose numerator is the differential of the quantity under the radical sign, is equal to twice that radical. Thus,

V. A most important process is that which is called Integration by parts; it depends on the following consideration, if y be a function of æ

hence it appears, that

d(ry) = xdy + ydx
:. xy = fxdy + fydz
Sxdy = xy-Sydx

Having resolved a differential into two factors, one of which can be imme

diately integrated, we may take this integral regarding the other factor as constant; we must then differentiate the result thus obtained, upon the supposition that the factor which we considered constant is the only variable, and then subtract the integral of this differential from the first result.

Thus, in order to obtain the integral of log. x dx, let us consider this differential as composed of two factors, log. z and dr. The integral of da is x; and therefore, considering log. ≈ as constant, the integral of log. x dx will be x log. z; now differentiate this result upon the supposition that log. z alone is variable, and we have x. ; subtract the integral of this differential from the

da

x

integral first obtained, and we shall have the whole integral required.

Numerous examples of the application of this principle will occur in what follows.

VI. From the operations performed in the differential calculus, we know by reverting the fundamental processes, that since