178 PRACTICAL MATHEMATICS 



101. The integration of sin w when n is an integer 

 (1) When n is odd, 



[sin 7 dQ = fsin 6 sin d0 



Put x = cos 0, then dx = sin dQ 



also sin 2 0=1- cos 2 = 1 - x 2 



Then fsin 7 d0 = -- f(l - a; 2 ) 3 dx 



- x 6 dx 



1 3 



= - cos 7 - cos 5 + cos 3 cos 

 7 5 



(2) When n is even, this method fails, for by putting sin w 

 = sin"" 1 sin and making the substitution x = cos 



fan" dQ = [sin 71 - 1 sin dQ 



f 



= - ( 



_ -r2\ 2 fj-r 



& ) mil 



and (1 x 2 ) is raised to a fractional power, since (n 1) is odd. 

 This will not give a definite expansion, but a series of an infinite 

 number of terms. It is necessary to work in an entirely different 

 manner and deal with the multiple angles of 0. 



To integrate sin 6 0. 



Now if x= cos + i sin 



- = cos i sin 



x 



and 2 cos = x + -, 2i sin = x 



X X 



Also x n = cos w0 + i sin w0 



1 



= cos ?x0 i sin w0 

 x n 



and 2 cos n0 = x n + , 2i sin w0 =# n - 



# n x* 



