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dx

by (6), (cos x)'

(cos x)3
= tan x + } (tan x)”.
dx

1

3 si

+

+ 3 log tan (cos x) (cos x)" 2 (cos)

2 dx (sin x)'

s(sin x)

+ (sin x)2} + log (sec x) by (3).

s da(sin a).

1

1

dx (sin x)

= COS X + sec X. (cos x) (cos x)" 1 s

{(cos x)? – 3

- 3 cos x} - X.
(sin x) 2 sin a

- 2 log (sin x).
(sin o)
S

1
{(sin x) – 3}
(cos x)"

(cos x) (sin x)

{(sin x)" - *}. (cos x) 8 5 (cosa)

do
s sin æ (cosa)

+ log (tan x).
2 (cos x):
du

1
*
(sin x)” (cos x)*

g cot 2 x.
3 sin x (cos x))

8 cos 2 x S )" )

+ (sin x)" (cos x)

(sin 2 x)3

sin 2 x da

1

+ log (tan x). (sin x)' cos a 4 (sina)* 2 (sin x)?

(17) If the function be (tan x)” the formula of reduction is

(tan x)"-1 sdx (tan x)"

- Sdw (tan x)"-?

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3

1

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- sdx

1 If the function be

the formula of reduction is

(tan x)"
dx

1
1

1
S

(tan x)" (n − 1) (tan x)-1 (tan x)*-21 Sdx (tan x)* = } (tan x)– tan x + x. sdx (tan x)' = } (tan x)® – 1 (tan x)* + } (tan x)2 + log (cos x). dx

1 (cot x)* + } (cot x) + log (sin x). (tan x)

S

(18) If the function be ma cos x, the formula of reduction is sdx x" cos x = x" sin x + nan-'cos x – n (n − 1) sd x x*-?cos X.

x. Sdx xé cos x = x2 sin x + 2x cos x 2 sin x. Sdx x cos x = av sin x + 3x? cos x 6x sin X – 6 cos x.

In the same way we find Sdx x sin x =– * cos x + sin x. sdx x* sin x=-2'cos x + 4x sin x + 12x* cos x – 24 x sin X – 24 cost.

(19) If the function be en* (cos x)" the formula of reduction is

sd x eo* (cos x)" €** (cos x)-' (a cos x+n sin x) n (n − 1)

Sdx e“ (cos x)*-?; a' + na

a” + na a similar formula exists for ea* (sin x)".

(a cos x + 2 sin x) 2 er Sdx @@* (cos x)' = “* cos x

a (a® + 4)

(a sin x-3 cos x), 6“ (a sin x-cosa) da ed* (sin 2)^= "" (sina)*

(a* +1) (a’ +9)

+

+

a? + 4

+

a* + 9

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a sin 7 x

7 cos 7 x Sdx * (sin x)" (cos x)

64

a+ 49 3(a sin 5x-5 cos 5x) a sin 3x-3 cos 33 5 (a sin x-cos

+ a* + 25

a? + 9

a® + 1

+

osa}

1

(20) If the function be

the formula of

(a + b cos x)"

reduction is

dx

Sta + 6 cos 2.)* = (n − 1) (a? – 6) (a + b cos 2.)*(n-1)(0-45/(a+b cos a ) -1 (n-1)763_069/(a+bcos x)****

-b sin x (n-2)

dx

(2n-3) a

dx

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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

dP dQ

dy dx should exist. When this criterion of integrability holds good, we find

d
U = (Pdx + sdy (Q [Pdx);

dy

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

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;

Integrating with respect to y,
u = by® + sdx {a +

(1 + c^) therefore the integral is

u = by® + ax + log C {x + (1 + x)}}.

wy dy yo dx (2) Let

= du. w? (x2 + y)2 Integrating with respect to y, and observing that there is no term in P involving y only, we find

(in2 + yo)?

U =

+ C.

P=

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dw dy wdy
(3) Let
+

:du,
(@? + y) y y (2* + y')
1

Q
(202 + y)2

y

(202 +

y) dP

y

dQ dy

(x2 + ya) dx
Since P does not contain any term independent of x,

d
sd v (P SQdy) = const.;

da
therefore, integrating with respect to y,

X + (x2 + y):
U = log y + log

+C;

y
whence U = log C {w + (x + y)"}.
(4) Let (ay + x3) dx + (13 + a x) dy = du.
The integral of this is

+ aʻxy + b3y + C.

4

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4

(5) Let (3xyw) dx – (1 + 6yo 3x*y) dy = du;

dP

dQ then

6xy dy

da

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