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Integrate me

  1. Mar 8, 2009 #1
    1. The problem statement, all variables and given/known data
    What is

    [tex]
    \int_0^{2 \pi} \; d\theta \sin^2 k\theta \cos^2 k\theta \; ?
    [/tex]


    2. Relevant equations
    Orthogonality of sines and cosines?


    3. The attempt at a solution
    I tried substitution and didn't get anywhere. Yeah, that's about it.
     
  2. jcsd
  3. Mar 8, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    Use the trig identity sin(2x)=2*sin(x)*cos(x) for a start.
     
  4. Mar 8, 2009 #3
    Whoa, I completely missed that. Using that identity, the integral becomes

    [tex]
    \begin{align*}
    & \int_0^{2\pi} d\theta \; \frac{1}{4} \sin^2 2k\theta\\
    &= \int_0^{2\pi} d\theta \; \frac{1}{8} (1 - \sin 4 k \theta )\\
    &= \frac{\pi}{4} \, ,
    \end{align*}
    [/tex]
    right?

    The answer just seems too simple--like there should be some k's around or something.
     
  5. Mar 8, 2009 #4

    Dick

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    Homework Helper

    You mean 1-cos(4k*theta), I hope. If k is an integer then there are no k's left around. If it isn't there are.
     
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