1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    User Avatar
    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

    User Avatar
    Science Advisor
    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integrate me
Loading...