INVERSE METHOD OF DEFINITE INTEGRALS. 327 



We shall now consider whether analogous formulee hold true for negative 

 values of n the index of differentiation. 



Generally if u and v be functions of t and fi'u denote the w"' suc- 

 cessive integral of ti, then 



for if we take the w* differential coefficient of each term in this series, 

 all the terms resulting mutually destroy each other except the first 

 term tiv. 



Putting u-=t'-"-^, v = t""-^, and rejecting the constants of integration 

 in the latter, we have 



also — - ^^L±lt'--l, ^ - (2« + l)(2« + 3) .,_„_! » 



Hence fiitf)'"-^ 

 (-2)"(^0-- (.,-. n 2«+l ,_^_, n{n-l) (2^+l)(2« + 3) „_ „ „ , 



= i.3.5....(2«-i)^^ "i-~T~-^ ^+-r¥-- Ts ^ ^-^^-'^ 



«r ^327 • dt-'^ ■ ^^^ > 



= cos {mcos-' (1 — 2^)^, 



the appendage which contains all the arbitrary constants being 



{^o + ^it + ^.f+...A„_J"-'\ . {tt')K 



