1877.] certain Definite Integrals. 361 



I have been asked to indicate the method of obtaining some of these 

 integrals. This I now do. 



(8.) is obtained by summing the series for by me ans of the defi- 



nite integral 



P log e cos cos 2rd = ( - 1Y- 1 . - . - . 

 1 4 r 



J o 



(9.) by expanding + - in terms of a, adding the resulting series 

 by means of the definite integral 



j: 



2 7 7T 1 



dd cos n d cos ^0 = - . — ;, 



o 



and combining the definite integrals arising from the process, by the rule 

 for the addition of binomial surds. To obtain (10.) we expand a ^ 



in terms of a, and then sum the series by means of the integral 



i 



2 dd cos n d cos nd = - 



J w dO cos ad 



2 2" 



(13.) is found in a similar manner. (14.) is obtained by using the integral 

 and (16.) by means of the integral which we used to ob- 



Jdx 

 by means 

 e*+l 



of the definite integral \ 6 smnd=( — l) n+1 -. (40.) is derived from 

 Jo n 



(19.) by integrating by parts ; but this integral can also be obtained by 

 expansion. (21.) and (22.) are obtained from the definite integral 



1 



cos n ~ l d dd cos {ctan0 + (n — 1)0} dd- 



(33.) is found by using the integral 



t 2 dd cos"- 1 sin (n+ 1) = — - . -. 

 J v ' 2»+ 1 n 



