we may obtain a series of the form 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.) + 1 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. nonS. **a*= n(n-1)x” – 2 &c x.log, a (w.log, a) (x.log, a)S + + 1.1 1.2.2 1.2.3.3 (nx) Si + &c. (m + 2)2 (m + 3)3 næ.log, a nx (nx) + + &c. (nw) &c. 1.2 m + 3 (m + 4)? ' (m + 5)3 log, a {.22 + &c. na xm + n.2 = 2m +1 {(m+ 1 1 1 + &c.} &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 c} - &c. n m+1 m +1 B + (m + 1) an 1 -20m+1 m +1 (n-1)(loge 2)n-1 ' (n-1)(n-2)(logex)"-2 log, (m + 1) + &c.} 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.) COS X =Scot w.cosecx= cosec x + const. (sin x) J, sin .cos 8=4(sin a) + cost. 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)". + S: (sin a)m+2. (cos )– 2; 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 + Vi (sin x)m+2 (cos x)"; n+1 m+1 + 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) n +1 n-1 n n n-1 + n n + n-1 n - 2 n=1 S= (sin a)" + *=S:700 (n-1)(sin x)" – 2 1 (n-1)(cos x)” – 1 (cos x)" – 2 (L. D. C. 250-8; L. C. D. 441-8.) a * (cos x)" (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 {w— n(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 Sx (cos x)" = sin a. Smm-1,-2 (cos x)" – 2m +1 n in n- 2 (sin x) – 1n – 2+1), (n even), m n m-1,-2_ (sin x) – In – 2+1), – In – 2an+1) + 1(n-1),2 log, tan fæ, (n odd). (n-1 2 m,-2 n in 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 m,-2 |