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we may obtain a series of the form
S, R (59) + S, R (92-) + &c.

R, (y)

++in which p-n~9-n may be made as small as we please : in either case, the value of the last term may be expressed by a converging series, by the methods already given.

(L. C. D. 423-6.)

+
Sy_R(2.) + Sy_(1+p2-92.) (1 +9*•92)*

1

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U=

- 1

. INTEGRATION OF TRANSCENDENTAL FUNCTIONS. (18.) Exponential and logarithmic functions. S. = logo x + const.

at

+ const. log, a

a*.dou a*.d; u S.a.u

+

&c.
log..a (log, a)? " (log,a)3
a

nonS. **a*=

n(n-1)x” – 2
+

&c
log, a (log: a)
at

x.log, a (w.log, a) (x.log, a)S
Semua
=log,« +

+

+ 1.1 1.2.2

1.2.3.3

(nx) Si

+

&c. (m + 2)2 (m + 3)3 næ.log, a

nx (nx) +

+

&c.
m + 2 (m + 3)? ' (m + 4)3
(næ.logx)

(nw)
+

&c. 1.2 m + 3 (m + 4)? ' (m + 5)3

log, a {.22

+ &c.

na

xm + n.2

= 2m +1

{(m+

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1

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&c.}

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&c.

S, u log,v=log, v.%.u-se, OSU.

, ulog, x=log,--" S: (log,«)" = (log.x)"). u-n (log, a)-° Season +n(n-1)(log,~)-° S Sada

d., UX .

(n − 1)(log x)" – 1 (n-1)(n-2)(log)" – 2

d (x.d,ux)

(n-1)(n-2)(n − 3)(log, x)" – 3 Secm(log.x)"= {(log,a)"

(logen)1
n(n = 1)
(log, x)" - ? - &c.

c}

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- &c.

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n

m+1

m +1

B

+

(m + 1)

an

1

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-20m+1

m +1 (n-1)(loge 2)n-1 ' (n-1)(n-2)(logex)"-2

log,

(m + 1)
(n-1)(n— 2)(n − 3)(log, «)n-3 + &c.

+

&c.}

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1

Salog, a

logo X (log 2) (log.x)3 =log, v + +

+

+&c. 1

1.2.2 1.2.3.3 (L. C. D. 427—37; Tr. L. 181–91; Hirsch, Int. Tab.)

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COS X

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=Scot w.cosecx= cosec x + const. (sin x) J, sin .cos 8=4(sin a) + cost.

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n-1

m+1

m +1

m

-1

+

n+1

m

-1

+

m+n

m+

(20) Formulæ for the reduction of Sx (sin x)". (cos w)".
Sit (sin x)". (cos w)" .
(sin x)"+1.(cos x)" – 1

+ S: (sin a)m+2. (cos )– 2;
(sin x)m-1. (cos x)"+1

n+iS. (sin @)m – 2. cos a)n +2 (sin x)”-1. (cos x)"+1

S. (sin a)m.–. (cosa)"; (sin x)"+1. (cos x)" – 1 n-1

+ S: (sin )". (cos x)" – 2;
m+n

m+n
(sin x)"+1. (cos x)"+1 m +n + 2

+ Vi (sin x)m+2 (cos x)"; n+1

m+1
(sin x)*+1.(cos x)" +1 m+n + 2

+

S. (sin x)". (cos X)*+2. n+1

(sin x)n-1.cos x Sx(sin x)". =

+ S (sin x)” – sin x (cos x)" – 1 Sz(cos x)" =

Se (cos æ)

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n +1

n-1

n

n

n-1 +

n

n

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+

n-1

n - 2

n=1 S= (sin a)" + *=S:700

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(n-1)(sin x)"

– 2

1 (n-1)(cos x)” – 1

(cos x)" – 2 (L. D. C. 250-8; L. C. D. 441-8.)

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a

* (cos x)"

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(21.) 8.2.209. sin x = - cos x {w" n(n-1)x1–2 + n... (n— 3)w? – 4 – &c.} + sin x {nw"?-n(n-1)(n − 2)– 3 + n... (n — 4)x1 – 5 – &c.}. Szc2.cosx=sin x {wn(n-1)– 2 + n... (n − 3)wn – 4 — &c.} + cos a {n 21–1–n(n − 1)(n 2)x1–3 + n... (n — 4) 21–5– &c.}.

Eat (sin x)".-1. (a sin & — n cos x) Sze** (sin x)" =

a+ na

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Sx (cos x)" = sin a. Smm-1,-2 (cos x)" – 2m +1

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n

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in

n- 2
COS X.
.S,, m-1,-2

(sin x) – 1n – 2+1), (n even),
-1

m

n

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m-1,-2_ (sin x) – In – 2+1),

– In – 2an+1) + 1(n-1),2 log, tan fæ, (n odd).

(n-1

2
(n-1),2

m,-2

n

in

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2 = sin æ.S,, m-1,-2

cos 20 – (n 2m+1), (n even), n - 1

S

m

m,-2

= sina. S -1,-2 cos x=(n=2m+2} + {n-1),2 log, cot (45° –4x), (n odd).

in=2 (n-1)

n-1

1
2
f(n-1),2

m,-2

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