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!

Integration proof

  1. Apr 15, 2013 #1
    I'm almost out of high school and have been trying to do some "harder" proofs, anyone, I'm not quite sure on how to proceed on this:

    prove that for all k:

    [tex] \displaystyle \int_0^{2\pi} (cos^{2k}x - sin^{2k} x) \ dx = 0 [/tex]

    If anyone could start me off as I'll I have in my head is "try to express it into something you can integrate", but having no luck.
     
    Last edited: Apr 15, 2013
  2. jcsd
  3. Apr 15, 2013 #2

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    It's not clear what k is limited to. Is k all integers, all reals, etc? If k is an integer, will its range include negative values? What about k = 0?
     
  4. Apr 15, 2013 #3
    The parts before this question asked me to integrate cos^6(x) and sin^6x using identities from cos^6(x) - sin^6(x) and cos^6x + sin^6x which I done successfully. At the end of the question it says: "You might like to consider how you would prove that [tex] \displaystyle \int_0^{2\pi} (cos^{2k}x - sin^{2k} x) \ dx = 0 [/tex] for all k, without having to derive a new identity for each value of k.
     
  5. Apr 15, 2013 #4

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    hint: use symmetry :wink:
     
  6. Apr 17, 2013 #5
    Could you expand on that? I tried rewriting cos in terms of sin, but still pretty stuck

    thanks
     
  7. Apr 17, 2013 #6

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    cos2k(π/2 - x) - sin2k(π/2 - x) = … ? :wink:
     
  8. Apr 17, 2013 #7
    = [itex] sin^{2k}x - cos^{2k}x = -(cos^{2k}x - sin^{2k} x) [/itex]

    I feel so dumb for not getting this :(

    I tried for around 15 minutes playing around with what you gave me to show it's 0 but to no avail... I'll try again tomorrow and post if I've made progress. Thanks for your patience.
     
  9. Apr 17, 2013 #8

    Curious3141

    User Avatar
    Homework Helper

    EDIT: Ignore my previous post, I'd made an error.
     
    Last edited: Apr 18, 2013
  10. Apr 17, 2013 #9

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Look at the graphs of sin(x) and cos(x). Can you get one of the graphs by shifting the other one to the right or the left?
     
  11. Apr 19, 2013 #10
    π
    yes cos(x) = sin(x + π/2)
    also sin(x) = cos(x - π/2)

    I can see why the integral is 0, by drawing both of the graphs and shading in the required area, but I'm having real trouble formalising a proof to actually put onto paper :\
     
  12. Apr 19, 2013 #11

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    try substituting y = π - x, or y = π/2 - x (and dy = -dx), and seeing what it does to the integral of one of your shaded parts :smile:
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Integration proof
  1. Integral Proofs (Replies: 1)

  2. Proof integrals (Replies: 5)

  3. Integration proof (Replies: 5)

  4. Integral proofing (Replies: 4)

  5. Integrability Proof (Replies: 10)

Loading...