The Rev. Brice Bronwin on some Definite Integrals. 495 



J - (sin*)" sm (m - 1) x = - 2> .p (n-1) . • (8.) 



h 3 

 Make w = 1, m = 2, 4, &c. ; multiply results by h, — , &c, 



3 



and sum 



/», /sin ,r\ , / 2 /* sin 2 a; \ , ^ 



dx {-—) tan-' ^ _ yiin . J = *tan-' -. . (9.) 



A 3 

 Let n and ;« be as before, and multiply by h, — — , &c, and 



9 



sum 



Let n — 2, w = 3, 5, &c. ; multiply by h, A 2 , &c, and sum 



/dx /sin x\ h sin 2 x sin 2 x ■■•» A . . 



~r~ V a- / ] — 2 A sin 2 # cos 2 .r + A 2 sin 4 x — ~2~ T+li * *' 



Examples here might be greatly multiplied. 

 From (7.) we obtain 



Integrating, and changing 2 into s 4- 1, m into m + 1, 



&„ - a.+ipff-i) A " {(»-^)"" ! ^(»-»») a > + c - 



But as n=m — 2 f, C is easily shown to be zero. Therefore 



/dx . 

 —r (s\n x) m sm % x 



= 2 "+'P(»-l) A " { ' Z -'")""' , ' Z - B ' )2} - J 



cos(z+l).r cos(z — l).r 



Since 2 cos # .r = s - h s * — , 



cos x cos .r 



2 / ~"77 (sin .r)™ cos * # = / — r (sin x) m ' '— 



J x n N *J x n cos # 



/^.r . cos (.?— l)x 



— (sn #) w — *„ ; 

 ^w V ' cos X 



or if 



pdx , . . COS2T.T _ - ,4.1 ~_f 



w* = / — (sin*)" , 2/ = if 1 ^ 1 + u % l . 



m > n J x n cos a; OT ' W m ' w w ' w 



Diminishing x by 2 successively till we arrive at 



2y- % = u- %+l +u- % ~\ 



J m,n m,n m,n ' 



