23, 24] DEFINITE INTEGRALS. 41 



If in the interval ab, (a<b),f(x) and <f>(x) are finite and inte- 

 grable functions, and \f(x) \ lies always between M and m , 



(5) 



f 



J a 



If in the interval <j> (x) always has the same sign, since 

 Mf(x) and f(x) m are positive, 



I* (M-fyj)<j>(x)dx and [ (/(a?) - m) < (a?) das, 



J a J a 



or 



Jf f % (a?) ^ - f /(*) < (a?) cfo 



J a J a 



rb rb 



and I /(a?) (f> (x)dx m I (f> (x) dx, 



J a J a 



rb 



have the same sign, and therefore f(x) </> (x) dx lies between 



J a 



rb rb rb 



Ml (j>(x)dx and ml <f>(x)dx so that I f (x) </>(%) dx is equal to 



J a J a J a 



rb 



I (j> (x) dx multiplied by a factor A lying between M and m. 



J a 



If /"(a?) is continuous, there is some point f in ab for which 

 /(f ) = A, and accordingly 



(6) f /(*) * (a?) ^ =/() f V () (to, a < f < b. 



J a J a 



This important theorem is known as du Bois-Reymond's 

 theorem of the mean. 



24. Indefinite Integrals. Let f(x) be integrable between 

 a and 6. The integral 



Cx 



I f(x) dx 



' a 



is zero for x = a, and for every value of x between a and b it has a 

 definite value. It is therefore a function of its upper limit x. 

 Let us denote it by F(x). If x + h be another value of x in ab, 



rx+h rx rx+Ji 



F(x + h)= /()*-//()*+;/ /(*)<to, 



^ a J a J x 



rx+h 



and ^ (x + h) - F(x) = /(a?) (to, 



y 



= /t^., M>A>m. 



