a Definite Integral. 



59 



Hence 



_£^ = - 1 . y fill 1 + J^±l 



j [ 2 



n ^ 



K = 



2 ZX COS 



2«4l 



2ra<>+l) 



2ic + l 



7r(«+l)— 2 COS TTCi . _ , _ 



w v ; n , (-l) a [l-(— 1)"] 



3? 2 — 2 A' COS 



2/f + l 



+ 1 



and 



2n(a? + l) ' 



. . . (13) 





cos 7r(a+lW log («r — 2,/' cos tt + 1) 



+ 2 sin 



-21o g ( 2 sin^±l.)} 



« i 2ra 



•n- — tan' 



2*+l "I 



''—COS 77 ^ 



sm 



7T ' .J 



+ (-l)ll-(-l)'] logK , +1) , 



(14) 



To find the Integral (10), we write 



m-l T> 



•c 1 



1 "A 

 L/ » . i \ = ^ '^ - * m = (p — B — l)>i — a (15) 



Multiplying both sides by #-r K and letting a?=r w we have 

 A = A^Z*1 = A_J_1 __ 1 r -, + i {u;) 



Multiplying both sides of (15) by x m , gives 



1 » A 0'»» »'• — 1 



(17) 



Differentiating both sides /c times and letting x = 0, we 

 obtain 



It follows that B =l, B 1 =B 2 = ....=B n _ 1 =0, B n =-1, 

 #2,1= +1, and in general B B(C =(— 1)\ 



B« = 



