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2 (sin x)2

+log (tan).

(tan a)" the formula of reduc

(tan x)"-1

- fdx (tan x)"-2.

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(n − 1) (tan ¿)^-1 − fdæ (tan x)=2

(tan æ)3 – tan æ + x.

(tan æ)ˆ − 4 (tan æ)* + 1⁄2 (tan x)2 + log (cos x).

=

-(cota)+(cot x) + log (sin x).

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

fdx x cos x = a" sin x + næ2-1 cos x

fdx x2 cos x = x2 sin x + 2x cos x fdx x3 cos x = x23 sin x + 3æ2 cos x

In the same way we find

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

2-2

n (n − 1) fdx x2-2 cos x.

2 sin x.

6x sin x 6 cos x.

fdæ æsina-x'cosa + 4a3sina +12a2 cosa - 24a sina - 24 cosø. (19) If the function be e" (cos a)" the formula of reduction is

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(2n-3) a

(n-2)

+

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dx

(n−1)(a2-b3) (a+b cos x)-1 (n−1)(a2-b2)√ (a+bcos x)" -2 √

Let n = 2, then

dx

(a + b cos x)2

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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

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

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therefore

d P

=

d Q 0 =

Integrating with respect to y,

u = by2 + fdx {a + (1 + 2)}};

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

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Since P does not contain any term independent of x,

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whence

x + (x2 + y2)}

y

+ C';

U = = log C {x + (x2 + y3)}}.

(4) Let (a3y + x3) dx + (b3 + a2x) dy

The integral of this is

= du.

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(5) Let (3xy x2) dx − (1 + 6y2 - 3x2y) dy = du;

1

then

dP

=

dQ

6xy=

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