30 DIFFERENTIAL AND INTEGRAL CALCULUS. 



if wz, n have the same sign, or 



1 ' 



log 



2 V(~ mn) D (V(- n) a - V 

 if n be negative and m positive, or vice versa. 

 The following integrals 



/x C x C 



"77~1 - 2\ dx. I -/TIT ~~2\ dx, \X m V(a' J - #' 2 ) dx, 

 \/(a a 8 ) J J V(2# - arj ' J 



and fx m J(2ax x z ) dx, 



frequently occur with different integral values of m, and 

 are to be obtained by what are called formulae of reduction ; 

 i.e. in each case the integral is expressed in terms of 

 another of the same form, but with a lower power of #, 

 and this is carried on until the integral involved is known.' 



L Take p 



and oT" a 2 - x* dx=x m ~* _ dx = 



therefore P m = - x m ~ l V( 2 - * 2 ) + (w - 1) ( 2 P WI _ 2 - PJ, 

 or W P W = - x m ~ l V(* - * 2 ) + (w - 1) o'P^. 



If the limits of the integral be and a ; or a, a, 

 the integrated part vanishes at both limits, and we have 

 x m _m-l T a x m ~ z dx 



II. Take P =- /1 - *dx 



- x) J V(2ic - x') 



and x m - l ^-l\/(2ax-x*}}dx 



^ rc" (2aic - ic) 



-^ - ( r i) J V(2aa; _^ J 



- x'} -(m-l) (-2nP m ., - PJ ; 



