CHAPTER X 



86. Integration. Integration is the converse of differentiation. 

 If we differentiate a certain function with respect to x, the effect 

 of integrating the result with respect to x will be to produce the 



original function. For example, if y = ax n , then -^ = nax n ~ v t 



ax 



and integrating nax n ~ l with respect to x will produce ax n . 

 The process of integration is denoted by the symbol I, and 



if x is the variable, the expression to be integrated is terminated 

 by dx. This at once distinguishes the variable from the con- 

 stants in the expression to be integrated. 



Thus \y dx means that y must be integrated with respect to x, 

 and this can be done provided we know the relation which gives 

 y in terms of x. Also \x dy means that x must be integrated 



with respect to y, and to do this we must know the relation which 

 gives x in terms of y. 



To integrate ax n , or to find \ax n dx. 



If y = ax m , then ^- = rfiax- 1 

 dx 



Since integration is the converse of differentiation, 



m - 1 dx = ax m 



f 

 \ 

 J 



Replacing m 1 by n, 



ax m 



or ax m - 1 dx = - 



m 



f ax n+l 



Then I ax n dx 



J 



n+ 1 



-Ttfl -\~L 



When the constant a = 1, then \x n dx = - This result 



J n+ 1 



holds for all values of the power n, except the case when n = .1. 



Cdx 

 To integrate x~ l or to find I 



J x 





