Integrating sin(2nx)/sin(x) for integer n

[Yegor]

Can you help me with
$$\int\frac{\sin(2nx)}{\sin(x)}dx$$
Here n=1,2,3...
I think that i should get any way to represent $$\sin(2nx)$$ as product of sinx and something. But i don't know how.
Thank you

 

[dextercioby]

Except for the integration constant,here's what Mathematica gives as an answer.

Daniel.

 

[Yegor]

Great. I have Mathematica too.
I'm given a hint. sin(2nx)=sin(x)*(Sum of trigonometric functions). I don't even understand how my head had to work to get such an idea.
How i had to think about this problem??

 

[dextercioby]

What are those equal to...?

$$\sin nx =...?$$

$$\cos nx =...?$$

in terms of the powers of "sin" and "cos" of "x"...?

Daniel.

 

[shmoe]

To write it as sin(x)*(Sum of trigonometric functions) you can replace you sines with exponentials, that is $$\sin(y)=(e^{iy}-e^{-iy})/(2i)$$. Things will factor, and you should be able to pull out a sum of cosines.

 

[Yegor]

I know only

$$\sin nx =\sin x \cos[(n-1)x] + \cos x \sin[(n-1)x]$$
$$\cos nx =\cos x \cos[(n-1)x] - \sin x \sin[(n-1)x]$$

These transformations can be maid also with $$\sin[(n-1)x]$$, and so on.
But how can i write that as a sum?