DIFFERENTIAL AND INTEGRAL CALCULUS. 21 



we conclude that C must for that particular value of m 

 have an infinite value, and we deduce the special case as 

 ro+i r wl+1 -1 1 



follows : fx m dx = +G = - - + C", since - - is also 

 m-i I m+l m + 1 



independent of x. Putting in this m = 1 , we have 



fdx , (x - l\ 



=0' + limit of (- 

 J x \m+l ; 



= C' + limit of : = 0'4 



\ u / tt=0 



(See the first chapter of Differential Calculus). 



X 



therefore a log a fa x dx. or fa x dx -, . 



log a 



These lead also to the integrals 



(x +'a) flHrl ( dx 

 (x + a)*dx = i -- ' , - - 

 w-fl J 



ax + b a 



Of course, as in differentiation, a constant multiplier 

 appears in the integral as a constant multiplier also, i.e. 



fa(j> (x) dx = a /$ (x) dx. 

 dx ri 1 1 



1 /$ I y-y\ 



This is if # <a ; if x> a. we should have logf - ) . 



2a & \x-aj 



(These two forms differ by the constant, although unreal, 

 quantity log (-1)). 



(5) -y- (sin aa?) = a cosaa?, fcosaxdx = - sin ax. 

 dx^ a 



(6) -y- (cosaa?) = a sinaa;, fsmaxdx = -- cosaa;. 



