The Rev. Brice Bronwin on some Definite Integrals. 493 



fa x fcE£ \ Z{l-2£sin 2 ^cos2x + £ 2 sm 4 #}=7rZ(l+^.(4..) 



k A 3 

 Make n = 2, m — 2, 4, &c. ; multiply results by — , — — , 



h b 



— , &c, and sum 



o 



/dx /sin.r\ ,/ hsmlx \ it , /2 + h\ ,- x 



T \nr) tan V i - y 8 in« J = ¥ * V^j- ( 5 -) 



We might derive others, and we may derive many more 

 from (b) very simply expressed, as 



/dx ( — l)i v 

 — (sin x) m cos (m - 2) x = ZOTTT, C*« ( 6 -) 

 x n v ■ v 7 2 w_n + 1 P(rc — 1) 



and from all thus derived we may find others by summa- 

 tion. 



If in (A.) we change f(x) into xf(x), we must not carry 

 the operations denoted by d and A so far by one step, or the 

 differential for z at the limit might not be zero. Thus the 

 second member of (A.) becomes 



iZJl £tn- 1 j d z n- 1 J d x sm x COS (z ~ VI + 1 ) xf{x) 



--~ & m - l J* n d z n ~ l Jd x sin {z-~m + 2) xf(x) 



.. '""^ A m ~ l f dz n ~ l fdxsin(z — m)xf(x) 



= (~^' A m /" dz n ~ l /dxsin(z— m)xf(x); 

 and (A.) becomes 

 / — (sin x) m cos (zx)f{x) 



i ,_i r ,(B,) 



= Lnil A m / d z n ~ l /dx sin (z — w) */(*) 

 2 W %J *J 



Here * sa t» — (2 i + 1). Make/(.r) = c~ 6 *, and 

 J dxc~ xx sin rx—^—s, Cdxsinrx =. — , 



pn-\ dr n-\ r n-2 ] 1 1 



But as A™r w-2 = 0, we may leave out the term containing A, 

 and we may change Ir into — lr*. Putting z — m for r, and 



