INVERSE METHOD OF DEFINITE INTEGRALS. 375 



i _ coeffi cient of sin30 in 2SiU _2\ 

 \ * ~ coefficient of sin29 in 2«S^m ~ 3 j 



S:i=2COS0S,-\Si 



= Po{cos39 — -cosO], .... 



J _ coefficient of sin 40 in 2S3U _ 5\ 

 \ ' ~ coefficient of sin30 in ZSaU ""6/ 



*S. = 2cos9S3 - Xs.S^ . 



= po|cos40 — -cos 20 — ->, 



f coefficient of sin 50 in 2SiU _ 91 

 \ '^ coefficient of sin4!0 in gAysM ~ lOj 



Si = 2cos9S^ - XiSi - 



= po |cos 50 - - cos 30 — -> . 



{2 2 11 



cos60 — ^cos40 — ^cos20 — ^> 



{2 2 2 1 



cos 70 — = COS 50 — - COS 30 — - cos 0> . 



Generally when n is an odd integer, suppose 

 -^^ = cos(w-l)0-^— {cos(m-3)0 + cos(« - 5)0 + ... + cos20 + i}, 



and — = cos »0 {cos (w — 2) + cos (« — 4) + ... + cos 30 + cos 0}. 



po n - - 



The coefficient of sin w0 in 2»S'„_,« = -. p„, 



n — 1 ^ 



of sin(M+l)0 in 2S„u = .p^; 



n 



therefore, x„ = <^±fi^ = 1 - ^ + ^ . 

 n(n + l) n n + 1 



