1

fdæ (sin x)3 (cos x)3 = − 1 (cos 10x - & cos 6a+ 5 cos 2 x).

fdx (sin x) (cos x)2

=

(cos 9x - cos 7 x +

=

+

by (6),

(cos x)1

3

(cos x)3

= tan x + (tan ∞)3.

dx

sin x

4

{(cos a)

+

+ log tan +

2 (cos x)2

(cos x)

dx (sin x)3

+ {(sin a) + (sin x)}+ log (see a) by (3).

14 2

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

fdx x cos x = x" sin x + nx2-1 cos x

fdx x2 cos x = x2 sin x + 2x cos x

fdx x3 cos x = x3 sin x + 3x2 cos x

In the same way we find

fdx x sin x=-x cos x + sin x.

n (n

1) fdx x2-2 cos x

6x sin x 6 cos x.

-

2 sin x.

fdxx sinx--x cos x + 4x3 sinx+12x2 cosx - 24 x sinx – 24 cos x. (19) If the function be e" (cos x)" the formula of reduction is

+

(n-2)

dx

(n−1)(a2-b2)J (a+b cosx)”-1 (n−1)(a2−b2) (a+bcos x)” −2 °

Let n =

dx

(a + b cos x)2

a2

1

b2

b sin x

2 a

+

tan-1

b

(a2 - b2)

a + b cos x

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 formula 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 x only, and in the latter those terms of P which involve y only.

therefore

0 =