Integration problem

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

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  • #2
dextercioby
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Except for the integration constant,here's what Mathematica gives as an answer.

Daniel.
 

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  • #3
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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??
 
  • #4
dextercioby
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What are those equal to...?

[tex] \sin nx =...? [/tex]

[tex] \cos nx =...? [/tex]

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

Daniel.
 
  • #5
shmoe
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To write it as sin(x)*(Sum of trigonometric functions) you can replace you sines with exponentials, that is [tex]\sin(y)=(e^{iy}-e^{-iy})/(2i)[/tex]. Things will factor, and you should be able to pull out a sum of cosines.
 
  • #6
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:blushing: I know only

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

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

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