INVERSE METHOD OF DEFINITE INTEGRALS. 381 



Ex. 4. Given ft(p{t) . {/{a + t) +f{a-t)\ = F{a), 



where the forms of the functions f and F are known, and that of 

 required. 



Put (f){t) = Co -1- Ci cos t + d cos %t + c-i cos St + kc. 

 f{a) = oo + a, cos« + 02 cos 2«+ "3 cos 3a + &c. 

 where a,„ a^, a^, &c. are known numerical quantities; hence 

 J'{a + t)+J^(a-t)=^2ao+2ai cosacos t+2a2 cos 2a cos 2l + 2a3 cos 3« cos 3^-r&c. 



and JP(«) = 27raoCo + 7raiCi COS« + wa^d COS2« + TrogCs cos3« + &c. ; 



therefore Co = — — - , c, = —^— . fa F{a) . cos « , c^ = -^— L F{a) cos 2«, &c. 



J w.. 1 r t:t/ X f 1 2C0S«C0S^ 2cOS2«COS2# „ ] 



and 7r(p{t) = — f„F{a) \—- + + + &c.} 



Tr [Zao Oj as J 



the hmits of all the integrals being and tr. 



Ex. 5. /■*%=J-,. 

 Jta — t a — h 



In this case we shall employ the functions V^ reciprocal to t". 



Put <^{t) = CflFo + Ci F", At C'^V-i + &c. «c? infinitum, 



1 1 !?;<'. , • ^ . 



and ;: = — \ — ; ^ — : + &c. «« infinitum; 



a-t a a' {^ "^ 



^, f 1 c„ 1 c, 2.1 6-2 3.2.1 Cs „ 



therefore r = ;r^ • -5 + ^ . e • ^ — ~. — ^ c <-, • -7 + &c- 



a-o a 2.3 e^ 3.4.5 a^ 4.5.6.7 «' 



1 * A= 6' 



= -+—,+— + —4 , &c. 

 a a^ ci^ a 



TJ 1 2.3 , 3.4.5 ,, 4.5.6.7 ,3 „ 



Hence Co = 1, c, = ~ . *, c, = ^ ^ . V, c^ = .^ ^ ^ . h\ &c. 



and <^{t) = r,-^.br, + ^^.b^F.,-^^^^.b^r.. &c. ■ 

 = r„ - 1. r,.(4i) + — •FAuy - 1^^. r3(4*r + &c. 



3d2 



