THE INTEGRATION OF sin" AND cos" 179 



Then (2i sin 0) -(a?- i 



- 20 + 15 -, - + " 



sin 6 = - (2cos 60-12 cos 40 + 30 cos 20-20} 

 64 



Therefore fsin 8 dQ = if (20 -30 cos 20 + 12 cos 40 -2 cos 60) dO 

 J 64 J 



- i( 209 - 15 sin 20+3 sin 40 - i sin 60) 

 o4l. 6 ) 



= -^{600 - 45 sin 20+9 sin 40 - sin 60} 



102. The integration of cos n when n is an integer 

 (1) When n is odd, 



f cos 5 d0 = [cos 4 cos d0 



Put x sin 0, then dx = cos dQ 



Also cos 2 0=1- sin 2 = 1 - x z 



Then f cos 5 d0 = f(l - a: 2 ) 2 dx 



2 1 



- - -iC 3 + -X* 



o o 



= sin - I sin 3 + i sin 6 

 3 5 



(2) When n is even, this method fails for the same reason that 

 it does in the case of sin n 0, but a result can be obtained by 

 working in terms of the multiple angles of 0. 



To integrate cos 4 



16 cos 4 = 



-r 



x* 



cos 4 - (2 cos 40 + 8 cos 20 + 6} 



