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Integration by parts: Reduction Formulas

  1. Sep 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Are there any sample problems worked out for trig functions of higher powers which are integrated by reduction formula? Like cos^8 or cos^10 or even just cos^6 or maybe?
     
  2. jcsd
  3. Sep 24, 2012 #2

    SammyS

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    You might try to "google" this.

    Wikipedia has a few examples worked. Follow the link: http://en.wikipedia.org/wiki/Integration_by_reduction_formulae#Examples
     
  4. Sep 24, 2012 #3

    Simon Bridge

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    Yep - or just work out the reduction formulas for arbitrary powers yourself ... good practice at integration by parts.
     
  5. Sep 25, 2012 #4
    Thanks SammyS and Simon Bridge
     
  6. Sep 25, 2012 #5

    Ray Vickson

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    For things like cos^n or sin^n the slickest way is to use Euler's formula. We have
    [tex] \cos(x) = \frac{1}{2}(e^{ix} + e^{-ix}), \text{ so}\\
    \cos^n(x) = \frac{1}{2^n} (e^{ix} + e^{-ix})^n
    = \frac{1}{2^n}\sum_{k=0}^n {n \choose k} e^{ix(2k-n)}\\
    = \frac{1}{2^n} \sum_{k=0}^n {n \choose k} \cos((2k-n)x).[/tex]
    You can get a similar result for ##\sin^n(x),## using the fact that
    [tex] \sin(x) = \frac{1}{2i}(e^{ix} - e^{-ix}).[/tex]
    For example, we get
    [tex] \cos^{10}(x) = \frac{63}{256}+\frac{105}{256} \cos(2x) +\frac{15}{64} \cos(4x)
    + \frac{45}{512} \cos(6x) + \frac{5}{256} \cos(8x) + \frac{1}{512} \cos(10x),[/tex]
    which is easy to integtrate.

    Things like sinp(x) *cosq(x) can be similarly expressed, although the expressions start to become lengthy and complicated.

    Unfortunately, this type of trick does not work for something like tann(x).

    RGV
     
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