Question

Give reduction formula for ∫ (sin nx/ sin x) dx.

Answer 1

Let In = ∫ (sin nx/ sin x) dx 

= ∫ ( 2cos(n – 1)x + (sin(n – 2)x)/ sin x) dx 

[ ∵ sin nx – sin(n – 2)x = 2cos(n – 1) x sin x 

(sin nx/ sinx) = 2cos(n – 1)x + (sin(n – 2)x)/ sin x) ]

(2 sin(n – 1)x)/ (n – 1)) + I_{n – 2} 

which is the required reduction formula.