(18) If the function be cos x, the formula of reduction is

fdxx sinx--x1 cos x + 4x3 sin x+12x2 cosx - 24x sinx - 24 cosx.

1

(20) If the function be

the formula of

(a + b cos x)"

dx

- b sin x

reduction is

√ (a + b cos x)2 = (n − 1) (a2 = b2) (a + b cos x)"−1

(2n-3) a

+

dx

-2

́ (n−1)(a2—b2)√ (a+b cos x)"1 ̄ ̄ (n−1)(a2 − b2) √ (a+b cos x)" - 2 '

Let n =

2, then

CHAPTER III.

INTEGRATION OF DIFFERENTIAL FUNCTIONS OF TWO OR MORE

VARIABLES.

SECT. 1.

Functions of the first order.

IN order that a differential function of two variables of the first order, such as

Pdx + Qdy,

should be the differential of a function u, it is necessary that the condition

The application of these formulæ may be generally facilitated by observing that in the second term of the former it is only necessary to integrate the terms in Q which involve a only, and in the latter those terms of P which involve y only.

dQ = 0 ==

Integrating with respect to y, and observing that there is no term in P involving y only, we find

Since P does not contain any term independent of x,

whence u = log C {x + (x2 + y2)}}.

(4) Let (a2y + x3) dx + (b3 + a2x) dy = du.

The integral of this is

(3xy-x) dx − (1 + 6y2 – 3x2y) dy = du;

then

6xy

=