CHAPTER XIII 



104. Integration by Parts. 



If y = uv 



du dv du 



then -j 2 - = w-j- -I- v-r- 



ax ax ax 



Integrating throughout with respect to x, 



f dv , f du , 

 uv = Iw-j- ax -} \v-r- ax 

 J dx J dx 



dv , [ du , 



or \u-j- ar = wt> IV-T- da; 



dx dc 



dw 



f 

 \ 

 J 



or symbolically IM dy = wu lu 



This rule enables us, in many cases, to integrate the product 

 of two different functions of x, for we can represent one of the 



functions by u and the other by dv in the integral IM dv. In 



order to build up the right-hand side we want the terms corre- 

 sponding to du and v. The first of these is obtained by dif- 

 ferentiating the function denoted by u, and the second by 

 integrating the function denoted by dv. It must not be assumed 

 that the application of the rule of integration by parts will enable 

 us to integrate a product straightway, for a consideration of 

 the right-hand side shows us that the method takes an integral 

 and splits it up into two parts, and the second part is an integral. 

 The success of the rule depends upon whether this second integral 

 is more easily dealt with than the original integral. 



The well-known integrals, to which the rule of integration by parts 

 can be successfully applied, can be divided into four distinct classes : 



(1) pMogjtfda: 



(2) y?* x n dx and p tl (log a ;r) m dx 



(3) \x n sin ax dx and la; n cos ax dx 



(4) 1 (?* sin bx dx and I e?* cos bx dx while with these can 

 be included J^ * sin n x dx and le * cos" x dx. 



183 



