44 DEFINITE INTEGRALS. [iNT. III. 



As x increases indefinitely, this approaches a finite limit only if 

 k > 1, when 



dx (a c) l ~ k 





a (x-C) k k-l 



26. Differentiation of a Definite Integral. Suppose that 

 the integrand is a function of a parameter u as well as of x. Then 

 in the case of a function of x that is capable of representation by a 

 curve, if we change the parameter u we change the curve, and if 

 f(x, u) is a continuous function of u, to an infinitesimal change in 

 u corresponds an infinitesimal change in the curve. The area 



rb 

 represented by the definite integral I f(x, u) dx changes by the 



J a 



area of the narrow strip added to or included between the two 

 curves, and we may find the ratio of this change to the given 

 change in u. We thus get a geometrical notion of the meaning of 

 the derivative of the integral with respect to u. Now by the 

 definition of the derivative 



rb rb 



. f(x, u + h) dx - f(x, u) dx 



Cu [ // v 7 -i' J a -J a 



j- f(x, u) dx = lim - r 



duj a j ^ ' h =o h 



h=Q 



It now becomes a question whether we may change the order 

 of taking the limits involved in the integration and in making h 

 approach zero. If f(x, u) is a continuous function of x and u we 

 may do this*, and since 



lim /(,+ A) -/(,> = a/ (*. u) 



h=o h du 



we have 



d f i , f b df(x,u) , 



j- f(x, u)du= ^-V - dx. 

 du] a j j a du 



We have already considered the definite integral as a function 

 of its upper limit, and have found, 24, 



d 



* So Kronecker, Theorie der einfachen und der vielfachen Integrate, p. 26, (the 

 word gleichmassig being superfluous, vid. Harkness and Morley, Theory of Functions, 

 64). For a more careful statement, see Tannery, Theorie des Fonctlons d'une 

 Variable, 166. 



