we may

obtain a series of the form

ƒ1R(y2) + ƒ1 R(y2,) + &c.

-n

in which p-q- 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.)

INTEGRATION OF TRANSCENDENTAL FUNCTIONS.

Exponential and logarithmic functions.

a*.u=

log.a

at

log, a

=log, x+

xm+n.x

=

xm+1

a*.d2u (log, a)

nx.log.x

1.1

1

-1

log, a

+

+1 1

(x.log, a) (x.log, a)3

+

&c.}

xm+nx.

1.2

α

∞

1.2.2

nx

nx

+

1.2.3.3

(m + 2)2 ' (m + 3)3

(m + 3)2

nx

(m + 4)2 ' (m + 5)3
(m + 5)3 − &c.)

(nx.log, x)" - 1

1

Su

1

ƒ ̧ u (log, a)" = (log, a)" ƒ ̧ u— n (log, a)" - 1ƒ. — S. u

x 00

u &c.

-1

(n − 1) (log, x)"−1

d2 (x.d2ux)

(n − 1) (n − 2) (log, x)” –›

(n − 1) (n − 2) (n − 3) (log, ∞)” − 3

xm+1

n

ƒ, a" (log, a)" === {(log, a)" — (log, x)" -1

(L. C. D. 427-37; Tr. L. 181-91; Hirsch, Int. Tab.)

(20) Formula for the reduction of ƒ (sin x)TM. (cos x)".

-2

- cos x {x" - n(n − 1) xn-2 + n... (n − 3) x2 - 4 — &c.}·

-1

+ sin x {n x2-1-n (n − 1) (n − 2) x2 - 3 + n... (n − 4) x2 -5 — &c.}.

-4

-

fxx". cos x = sin x {x” — n (n − 1) x2 − 2+n.....(n· ...(n − 3) x2 - 4 — &c.}

-3

+ cos x {nxn- — 1 — n (n − 1) (n − 2) x2 − 3 + n... (n − 4) x” —5— &c.}.

n

in

n

1

cos x. Sm-1,-2 (sin x) − (n − 2m+1), (n even),

m,-2

m,-2

(sin x) − (n − 2m+ 1) + (-1),2log, tan, (n odd).

2

(n-1),2

COS x ̄(n−2m+1) + 1 (n−1),2 log, cot (45°—1x), (n odd).

2
{(n-1),2