INVERSE METHOD OF DEFINITE INTEGRALS. 185 



SECTION V. 



Inverse Method for Definite Integrals which are expressed in positive 

 powers of x, or under any form. 



18. Let <^{x) represent any function of x, such that Stf(Jt) .f = (p{x) 

 when X is any integer from to n — 1 inclusive, then excluding the 

 case of (p {x) = 0, which has been considered in the preceding Section, 

 it is evident that by putting 



f{t) = A, + A,t + A,f + +An-,tf-\ 



the conditions of the question give n equations, which suffice to de- 

 termine the coefficients A^, Ai, A^, A„.^\ if we represent the 



particular value of f{t) thus deduced by T„^i, and seek its most 

 general value, we have 



;/(0 .t^ = <p {x), 



.-. f,{f(t)-T„.,}.t^ = 0. 

 Hence by the preceding Section, the most general value of f{t)— Tn-i is 



dt- ' 



and therefore the most general value oi f(t) is found by adding this 

 appendage to its prime value T„_i. 



19. When <p{x) is a rational and entire function of x, of m di- 

 mensions, we have by the proposed conditions 



'P^'^'' x+l^ x + 2^ x + 3^ x + n' 



and actually adding the terms which compose the right-hand member 

 of this equation, the common denominator is 



(x + l){x + 2) (x + n), 



s2 



