INVERSE METHOD OF DEFINITE INTEGRALS. 355 



Multiply by 2A"+i, and diiFerentiating once more, we get 



^Thr djf^ 1 



= 1.2.3...2wx(2« + l)P„A'-i + 2.3...(2w + l)x(2» + 3)P„+,A"+J+&c. 



Hence, F„ = ^h-^^ . j. [jf^l ^^ j^ I "^ ' " "^> -^,f ^ . 



c?A\ 1.2.3...(2« + 1).</A^° j' 



when A is put = 1. 



Put for abridgment the radical {1 — 2A (1 - 2#) -f A^}-J = ^, then 



rf^ . {Rh") _ 2«.(2w-l)...(w + l) ^^B 



rfA^ ~ 1.2...« • • </A'' 



. 2w(2w- l)...w ^ ^ ,c?"+'J? . 

 ^ ' ' ' \c?A" W + 1 1 C?>&»+1 



w(w-l) _A^ c?»+^B 1 



(« + l)(« + 2)'1.2'rfA»*'" *''^7 



Whence 2 ~ {^-i ^^^^| = 2» (2«- !)...(« + 1) {(2« + 1) A«-4 ^ 



2w + 3 n ,^^ d'^'R 2n + 5 h"*^ .n .{n-l) d'^^R 

 ■^ 1 •« + ! • dh"^' "^ 1 . 2 * (» + l) (w + 2) •• d¥^ ■•■ *'*'• 



c?A"+' w + 1 c?A"+- J 



u 100 Tr d'R . 2w + 3 « .d^^'R 

 Hence 1 .2.3.. .wF^, = -jT- + r — -^. .^ „ ,, 



2w + 5 w (w-1) ^^ rf»^^jB 



"'' 2» + 1 ■ (» + 1) (ra + 2) ■ 1 . 2 • rfF^ "*■ *'''• 



2 ^^ c?"^^Jg 2 n d'^'R 



2« + 1 * t/A"+' "^ 2w + l» + l ■ </A"+^ 



2 n(n-l) h^ d'^^R 



2w + l*(w + l)(« + 2)*1.2' <^A»^' ■•" *'^- 

 Vol. V. Part HI. 3 A 



