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!

Homework Help: Integeal of Sin^2

  1. Mar 28, 2012 #1
    Hi guys,

    I have a problem with the following integrals( the integral is between [-L,L]) :

    1) ∫sin(( nπ/L )x).sin(( mπ/L )x) dx = L when m = n

    and :

    2) ∫sin(( nπ/L )x).sin(( mπ/L )x) dx= 0 when m ≠ n

    I know that in the first equation we have :

    ∫sin^2(ax) dx = (x/2) - (sin2ax/4a)

    but it doesn't work like that, could you please help me with it?

    and for the second equation I think when m≠n then m or n should be even and one of the sin will be 0 so the integral will be 0, as in second equation. Am I right about it?

    Thanks in advance.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Mar 28, 2012 #2
    What do you mean it doesn't work like that? Isn't that completely correct?

    It should work for any m and n -- perhaps integrating in parts might help.
  4. Mar 28, 2012 #3


    User Avatar
    Homework Helper
    Gold Member

    Yes, i was going to say, it does work like that.

    This exercise is meant to introduce you to a very major mathematical theme.

    Apart from calculations, for the case m ≠ n do a sketch of sin(( nπ/L )x) and sin(( mπ/L )x) on same fig. over nm cycles*. e.g. n = 1, m = 2 or n = 2, m = 3. Look at any value of sin(( nπ/L )x) - there are general several. Look what the value of sin(( mπ/L )x) is at that x. For every value some you will find that at some of your x points sin(( mπ/L )x) is exactly minus what it is at other such points. If you think about it you may realise this has to be. For every point. Result: this product, and hence your definite integral averages out as 0. On the other hand when n = m you have a square which is never negative so the integral is not 0 in this one case.

    Very useful conclusion you will hear no end of later.

    *or lcm of n, m
    Last edited: Mar 28, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook