24, 25] DEFINITE INTEGRALS. 43 



the true nature of the definite integral being that of the limit of a 

 sum. 



25. Infinite Integrand or Limit. The definition of the 

 definite integral presupposed that the integrand was finite in the 

 field of integration ab. If there should exist points in the region 

 ab at which f(x) became infinite, the integral would in general 

 have no meaning. In case however there is a single point c for 



which f(x) becomes infinite, if h^ and h 2 are positive numbers 



rc-hi rb 



however small, the integrals I f(x)dx, /(#) das have a de- 



Ja J c+h 2 



finite meaning. If now as h^ and h 2 approach zero independently 

 of each other the sum 



-&i c b 



f(x)dx+ f(x)dx, 

 J C+hz 



approaches a definite finite limit, the value of that limit is what is 

 meant by the definite integral, 



f /<*)* 



J a 



For example, let 



then for x = c, f(x) becomes infinite. 



f 6 das f c - fe > dx f 6 das 



7 -- \fc = l im 7 - 0- + ^ lm / - \* 



Ja (X- C)" 7tj=0 J a . (X-c) k h2 =Q J c +n, (* - C>* 



r (- h^-* -(a- c)*- k + (b- c)*- k - (h^~ k 



l-k 



There is a limit as Aj and h 2 approach zero only if 1 k > 0. 



rx 



In like manner if the integral I f(x) dx approaches a finite 



J a 



limit when the limit x increases indefinitely, then this value 

 defines the meaning of the definite integral 



["/(*) d* 



J a 



Let, as before, 



r ^ 



Ja^ 



das 



C) k 



