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Simple Integral

  1. Oct 14, 2011 #1
    1. The problem statement, all variables and given/known data

    [tex]\int sin(101x) sin^99(x) dx [/tex]

    2. Relevant equations

    Complex Number

    3. The attempt at a solution

    [tex] sin(101x) = \frac{e^{101ix}-e^{-101ix}}{2i} [/tex]
    [tex] sin^99(x) = Im(e^{99ix}) [/tex]

    Still trying...
     
    Last edited by a moderator: Oct 15, 2011
  2. jcsd
  3. Oct 14, 2011 #2
    [tex]\sin x=\frac{e^{ix}-e^{-ix}}{2i}[/tex]
    Does that help?

    edit: Ah, sorry. Didn't see the mangled tex. just a minute.
     
  4. Oct 14, 2011 #3
    There is an identity for [tex]sin^nx[/tex] which transforms it into a sum of regular sines. Perhaps that is a place to start.
     
  5. Oct 15, 2011 #4

    lurflurf

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    Homework Helper

    use reduction formulae
    try an identity from elementary trigonometry such as
    sin(101x)sin(9x)^9=[exp(101 i x)-exp(-101 i x)][exp(9 i x)-exp(-9 i x)]^9/2^10
    from which (or otherwise) one may see that
    sin(101x)sin(9x)^9=(1/512)(cos(20 x)-9 cos(38 x)+36 cos(56 x)-84 cos(74 x)+126 cos(92 x)-126 cos(110 x)+84 cos(128 x)-36 cos(146 x)+9 cos(164 x)-cos(182 x))
     
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