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16** (cos x)" =

€4* (cos x)" – (a cos a + n sin x)

a + n?
n(n-1)
+

"
’ ?

(a sin ba b cos bx) Steam. sin bx=

a +62

(a cos bx + b sin ) Szeax.cosbx=

a + 6 From these formulæ Ste*. (sin bx)". (cos cæ)" may be obtained, if (sin bx)" and (cos cx)" be expanded in series of multiple sines and cosines.

ar

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1

=

(62 — a?)] logo

(b + a)(1 + cos x)| + (b- a)(1 – Cos x)

+ const., 13 (b + a)(1 + cos x) (b-a)(1-сos x)

3 1

6+ a cos x + sin x (b? a?): log.

+ const. if a <b: (6-a')

a + b cos x

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Statb cosa Si a + b cos &

COS X

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Sza+bcos

a + b cos x b

x a + bcos a
S.
a, + 6, cos x

(ab, - a;b) sin a
(a + b cos x)" (n-1)(a? b)(a + b cos x)" = ?
1

(n-1)(aa, bb)+(n 2)(ab, - a,b) cos x (n-1)(a?–63) S.

(a + b cos x)" -1

+

e +

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2

4

m 1

+ 2

el

2

m...

+

&c.}

S:(1+e.cosx)" = AX + A, sin x +{A, sin 2x + }A, sin 3x + &c.

mm - 1) 1 m...(m— 3) 1.3 A =1+

-e* + &c. 1.

1...

2.4
m (m - 1)(m-2) 1.3
A=2e

1.

3 2.4 (m— 4) 1.3.5

et + &c. 1... 5

2.4.6 2me A - 24

(m - 1)e A, -23 A, A,

A2=
(m +2)

(m + 3) e
(m 2)e A, — 23 Az
A4 =

&c. = &c. (m + 4) e If / (1+e.cos x) –m= AX + A, sin x + A, sin 2 x + &c. and S (1-e.cos x) –m-l=Bx + B, sin x + 1 B, sin 2x + &c.,

2mA- (m-1) A, then B=

= A + = d.A, 2m (1 - e)

e

т

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e

2 k J, log. (1 +e.cos a)= -a

= – log: + 2 {k.cos x 1ko.cos 2x + }k.cos 38 – &c.} where k=-{1-(1–e®)}}.

(L. C. D. 449—66.)

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APPROXIMATE VALUES OF INTEGRALS. (23.) Let Sr=q u represent the value of Szu, when x=a; then

62 Scu=(u).a + de quo + du. This series does not converge with sufficient rapidity, unless b is very small : the following more convergent series

+ &c.

1.2

1.2.3

may however be obtained, by dividing the difference between the limits into n equal intervals ;

b Soatno

u, 72

{dozau + deza+bu + &c. + drzat(n-1)U}

+

1.2

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= uc

{(u),zato + (u).=a+26+ &c. + (u), atno}

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i {(u)

{(u)s=a+b

+

6

r=ato + &c. + (U).=a+(n–138 +1.(u)-a + (u)atno} 62 1.2.5 {d.cg U — deza+no

} + &c. The values of Szu between the limits x=a, and x=a+nb determined from the first series are always less, and from the second, alternately greater and less than the true value. The third series, which is half the sum of the two first, is more convenient for calculation.

This method is inapplicable, if any value of w between the assigned limits renders any of the differential coefficients infinite.

(L. C. D. 467–73; TT. L. 209–13.)

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SUCCESSIVE INTEGRATION. (24.) Szu=x/zu- Szux. Su «.

1.2 u

3x3wS 1.2.3 &c. = &c.

-1

XN
1.2...(n − 1)
(n − 1)(n-2)

&c. +
1 .
(25.) Series independent of the integral sign.

X? Su=

dut d u— &c. +2,00 + Cą. 1.3

1.2.4

-3

+

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2

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} +

.

1

X3

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U=

Siu

d,u +

&c. + x + x +cz: 1.2 (3 1.4

1.2.5 &c. = &c. x"

Q? SE"?

d, ut 1.2...(n-1) in 1(n+1)

) + C20"-1 + 0,20" 2 + &c. + cm; C1, C, &c. being arbitrary functions. (L. C. D. 483_5.)

u=

B

INTEGRATION OF FUNCTIONS OF SEVERAL VARIABLES.

(26.) If u is a function of x and y, then

u=Sxdu + (y), an arbitrary function of y, which would have disappeared by differentiation, being added instead of an arbitrary constant.

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If d.d,u=v, then d,u=/yv+f(x), and

u=S:8,0 +0(x) +0:(). Similarly if d,d, d,u=v, then

u='+/y8,0 +0.(x) + 2(9) +03(x); and so on, whatever may be the number of variables.

(L. D. C. 279—81.)

DIFFERENTIAL EQUATIONS. (27.) If any equation p(x,y)=0 contains n independent constants, then

[1] n differential equations of the first order may be obtained, from each of which one of the n constants has disappeared:

[2] any one of these n equations will be of the same degree with respect to the differential coefficient, as the highest power of the constant which has disappeared :

m (m - 1)(m - 2)...(m - m +1) [3]' there will be

differen2 . 3 tial equations of the mth order, each involving n-m of the n constants :

[4] there will be but one differential equation of the neth order, which contains no constant. From

any

differential equation of the mth order m equations of the (m— 1)th order may be obtained, in each of which one constant has been introduced.

(L. D. C. 287-304.)

1.

m

(28.) The criterion of integrability. If the equation

M+Nd.y=0 can be obtained by differentiation alone from the equation U=0, then

d, M=d,N, and

u=S,N+Sx(M-dz/,N). Szd,M=dyS&M; by means of which either of the coefficients M, N, may be obtained from the other.

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