dy=3ax dx

y=f3ax* dx

= ax3 + C

where C may be either positive, negative, or 0. We cannot determine 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, if

II. Since, if

it is manifest that

y=f4ax3 dx + 3bx* dx — S2 bx dx +ƒ dx + C

y=azm

dy mazmi dz

The integral of a function raised to any power is obtained 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 azm dz. Thus,

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

Let

√2ax

= √2a

y=

dx

(2a - x)2 + C

dy = (x2 + y2)* (6xa dx + 4y !!y)

dz

x3 + y2 = z .: 3x2 dx + 2y dy = dz.. 6x dx + 4y dy = 2dz

(x3 + y2)2 (6xo dx + 4y dy) = 2z3 dz

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

Sz-1dz = 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 x

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. x and dr. The integral of dæ is x; and therefore, considering log. x as constant, the integral of log. x dx will be x log. a; now differentiate this result upon the supposition that log. ≈ alone is dx

variable, and we have a.; subtract the integral of this differential from the

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

And in a similar manner, from the chapter on Inverse Functions, we have

b sin. x

-dx

-1

a

cos.x + C

=

2

a

In all these integrals the radius of the arcs is unity, and the arbitrary constant is not annexed to the integrals in the right hand column, for want of breadth of page. As it is frequently desirable to integrate differentials in which the radius is a instead of unity, we shall exhibit a few of those which most frequently occur to that radius. In the left hand column of the above differentials, write for X, and we have to radius a.

a

These are the elementary forms to which every differential whose integral is required must be decomposed; and the reduction of expressions to one or more of these fundamental formulæ is the object of almost every process in the Integral Calculus. The following are a few examples.