492 The Rev. Brice Bronwin on some Definite Integrals. 



/» edr n r n-\ r 

 T~, — s = d7 TT tan h e F (r, e). 

 s 2 + r l P(n — 1) s v ' 



To abridge, let tan -1 — == 6 r . If t = 0, 0,. = — - or — , 



according as r is positive or negative. Putting therefore z — m 

 for r, we have 



/ ( [ x i |.y 



B\xt^A m {z-m) n ~ l = 0; if we add it to ^ m {(z-m) n 'H Zmmtn } 

 it will destroy the terms where 9 is negative, and give 



^{{z- m y-H z _ n } 



=J(z + m) n - l -™(z + m-2) n ' l + m (™~% + m-4) H - l -bc\(a.) 



the series continued till we arrive at powers of negative quan- 

 tities, which is always to be understood of similar series. 

 Let if denote the first member of (1.). It is evident that 



J m,n v ' 



when z = or > m, or when z = or z_ (—m). if = 0. This 



function is of such a nature that we may change s into — z 

 without altering its value. Thus instead of (a.),, we may 

 employ, for the most part with great advantage, 



A m {(z-m) n - 1 $ z _ m } =7rj(m-z) K - 1 -^(w-z-2)"" 1 + &c.i. . {b.) 



In fact we might have subtracted — A m (z — m) n ~ l = instead 



ofaddingit,and reserved only the terms where is negative; and 

 since m and n are both odd or both even, we may change the 

 last term of A m {(z — m) n ~ l 8 z _ m }, which is ±{z—m) n - 1 Q % _ m 

 into + (m— z) n ~ l Q m -. zi and the preceding ones in like manner. 

 Hence we obtain (b.). This will give immediately 



/ -^ (s\n x) m cos (m - 1) x = ^~ » • • (2.) 



' x n v ' v ; 2 m V(n — 1) 



Make in this last m and n = 1, 2, 3, &c; multiply the results 

 by 1, /z, h% &c, and summing to infinity, we have 



//sin x\ q* — h*sm*x _ (A 1 \ 



^V~/^-/^sin2* + A 2 sin 2 x- 7r V 2 " ¥/' (3,) 

 Makew=l, ro=l, 3, 5, &c. ; multiply the results by 1, 



— , — , &c, and sum 



