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Integration problem

  1. Jun 14, 2005 #1
    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
     
  2. jcsd
  3. Jun 14, 2005 #2

    dextercioby

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

    Daniel.
     

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  4. Jun 14, 2005 #3
    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??
     
  5. Jun 14, 2005 #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.
     
  6. Jun 14, 2005 #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.
     
  7. Jun 14, 2005 #6
    :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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