180 PRACTICAL MATHEMATICS 



Therefore [ cos 4 d0 = -?-[(2 cos 40+8 cos 20 + 6) dQ 



- ^ (s sin 40+4 sin 20 + 60) 

 16 \2 / 



^{sin 40 + 8 sin 20 + 120} 



103. The integration of sin n cos m where n and m are integers. 

 The substitutions x = sin 0, or x = cos will enable us to inte- 

 grate this expression, except the case when n and m are both 

 even. 



cos 3 dQ = [sin 3 cos 2 cos 



(1) [sin 3 cos 3 d0 = [si 



(2) [sin 4 



= \x 3 (l x 2 ) dx, when x = sin 



"If 7. If 



= - sin 4 - sin 6 

 4 6 



cos 3 d0 = sin 4 cos 2 cos dQ 



= \x 4 (l x 2 ) dx, when x = sin 



1 7 



- 



- sin 5 - sin 7 

 5 7 



sin 3 cos 4 dQ = sin 2 cos 4 sin 



(3) [sin 3 cos 4 dQ = [si 



\x 4 (l x 2 ) dx, when x = cos 



1 7 1 ^6 



= 7* -5^ 



= - cos 7 - cos 5 

 7 5 



(4) When n and m are both even we must use the application 

 of De Moivre's Theorem. 



