INVERSE METHOD OF DEFINITE INTEGRALS. 351 



From whence we obtain 



A„^, = 2n + 3, A„^,= ^ ^ .(2n + 5), Jn+3=~ fgg '.(2n + 7), 



and to prove that this law of formation is general, we may observe 

 that since 



/ iy+2^+i_ 2n + 2x+l 1 (2w + 2a; + 1) (2n + 2x)...{x + 1) 



V~ h) ~ 1 ' h^ "^ 1.2...(2w + ^ + l) 



iy+^-^' I X 1 x.jx-l) i,^\ 



^ \ hi '\ 2n + x + 2' h {2n + x + 2){2n + x + S)' K" ]' 



Qfi 4- 2 

 and (l-hy^'-^'Hl + h) = l + {2n+.3) . h + . {2n + 5).k' 



(2« + 2) (2w + 3) ,„ „. ,3 J 

 + ^ o 9 • ^^^ + 7).h^ + &c. 



Multiply both, and take the coefficient of ,^„^^^, in the products, 

 and we get 



{2n + 2x + ]) {2n + 2x),..{x + 1) ^ x , 



\.2.3...{2n+x + \) * ~ 2n + x + 2'^ ' 



x{x-l) (2w + 2)(2w + 5) 



■*"(2w + a; + 2) (2« + a; + 3)' 1.2 ' ^^ 



= coefficient of -1- in (-IV (l+^Xl-^r'' 



= (-1)'. coefficient of A' in (1 +^) (1-^)^'-. 



Now the coefficient of h" in (1 +^) (1 -Af""', is evidently the sum 

 of the coefficients of h'~\ and of h\ in the expansion of (1 — A)*'-'; 

 that is, the sum of the coefficients of the two middle terms in a 

 binomial raised to an odd power, and with alternate signs of + and 



