of Definite Integrals. 437 



and change a into oc, we shall have the function itself by dividing 

 2 S by tt/i y/ — I, but when the above series is divergent, it be- 

 comes troublesome to calculate the Analytical values of the 

 divergent parts, to remove which difficulty we should attend to 

 the values of the divergent series arising from differentiatini: 

 successively the equation 



- = cos 6 — l cos 39+1 cos 5 9, &c. 



thus should be substituted for i - .i +5 - 7, &c. 



for 1"- 3 + 5'- 7 , &c. 

 &c. 



Definite integrals may be applied to the expansion of functions 

 when they contain negative powers of x, and they serve to de- 

 termine ihe coefficients of such terms. 



Thus, if we wish to find P, the coefficient of 1'' in /(«), 

 when such a term enters, we have 



■t!^ = P + Qj:' + Rxf + &c. 



xi' 



+ q.v'"-\- rx''' + &c. 



the upper line containing the positive, and the lower the negative, 

 powers. 



Multiply both sides by ^~, and integrate from .1= - 1 to a= + 1. 

 Hence, 



2 '2r I 



i - -r — &c. 



a I) J 



supposing a, /3, a, b, &c. to be odd, for the terms containing even 

 powers entirely disappear in the integration. 



