Can You Prove the Reduction Formula for Integral of Sin(nx) Over Sin(x)?

[e179285]

If I _n =integral sin (nx) dx/sinx,prove that I_n =2sin((n-1)x)/n-1 +I_n-2 for all integers n>(egual) .


I don't know how to start...

 

[HallsofIvy]

Since you are given the formula, the obvious approach is to use induction.

 

[SammyS]

If I _n =integral sin (nx) dx/sinx,prove that I_n =2sin((n-1)x)/n-1 +I_n-2 for all integers n>(equal) .

I don't know how to start...

sin(nx) = sin((n-1)x+x)

=sin((n-1)x)∙cos(x) + cos((n-1)x)∙sin(x)​

is a good place to start.