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Integration of Cosine

  1. Jan 8, 2013 #1
    1. The problem statement, all variables and given/known data

    [itex]\int { { cos }^{ 2n+1 }(x)dx } [/itex]

    2. Relevant equations
    [itex]{ cos }^{ 2 }+{ sin }^{ 2 } = 1 [/itex]

    3. The attempt at a solution
    i got to here:
    [itex]\int { { (1-{ sin }^{ 2 }(x)) }^{ n }d(sin(x)) } [/itex]

    Any help would be appreciated!
     
  2. jcsd
  3. Jan 8, 2013 #2

    micromass

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    So you got to find

    [tex]\int (1-x^2)^ndx.[/tex]

    Use the binomial theorem and linearity of the integral.
     
  4. Jan 8, 2013 #3

    Ray Vickson

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    There is no simple formula for the result, unless you want to express it in terms of a hypergeometric function. The answer is a polynomial of degree (2n+1) in ##\sin(x)##.

    BTW: in [t e x] you should write "\sin(x)" instead of "sin(x)": the difference is ##\sin(x)## (nice) vs. ##sin(x)## (not nice).
     
  5. Jan 8, 2013 #4
    Thanks for the tex hint..still getting the hang of it (used to microsoft equation editor =\)

    Thanks, i actually read about it, thought that i should use it but decided not to.

    In case anyone wants the answer, it is:
    [tex]\int { { (1-{ \sin }^{ 2 }(x)) }^{ n }d(\sin (x)) } =\sin (x)-\frac { n{ \sin }^{ 3 }(x) }{ 3 } +...+\frac { { (-1) }^{ n }{ \sin }^{ 2n+1 }(x) }{ 2n+1 } +C[/tex]
     
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