186 PRACTICAL MATHEMATICS 



106. (3) To integrate x n sin ax and x n cos ax. From the previous 

 example it is obvious that by putting u = x n the integral on 

 the right-hand side will contain x n ~ l , that is, by applying the 

 rule once we diminish the power of x by unity in the result- 

 ing integral. Hence to completely evaluate I x n sin ax dx or 



\x n cos ax dx, we should have to apply the rule n times. We 



can make the work more methodical by applying the rule once 

 to each of these integrals and using the results as standard forms 

 for integral values of the power n. 



Now \x n sin ax dx --= uv \v du 



where u = x n , du = nx n ~ l dx 



f 1 



and dv = sin ax dx, v I sin ax dx = cos ax 



a 



f 1 nf 



Then \x n sin ax dx = x n cos ax H la;"" 1 cos ax dx ... (1) 



J a a] 



Also \x n cos ax dx = uv \v du 



where u = x n , du = nx n ~ l dx 



f 1 



and dv = cos ax dx, v = Icos ax dx = - sin ax 



J 



Then \x n cosax dx = - x n sin ax \x n ~ l sin ax dx . . . (2) 



J J 



As an example, let us apply these results to integrate x 5 sin 2x. 



f 1 n f 



Then \x n sin 2x dx = - x n cos 2x + - \atP~ 1 cos 2x dx 



and \x n cos 2a; dx = - x n sin 2x - \x n ~ l sin 2x dx 



J' 



5 sin 2x dx 



1 K 5, 



= -x 5 cos 2a; + - |# 4 cos 



a 2i 



x 5 5 fl f 1 



= cos 2x + - \ -x 4 sin 2x 2 \x 3 sin 2x dx V 



2i \. J J 



x 5 5a? 4 f 1 3f 1 



= cos 2# + ' sin 2x 5 \ -x 3 cos 2x + -\x z cos 2xdx\ 



2 4 I. 2 2J J 



