1. Not finding help here? Sign up for a free 30min 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!

A sinusoid integrated from -infinity to infinity

  1. Apr 2, 2007 #1
    I had a sort of odd question on my homework,

    Sin(x)^3 dx, integrated over all reals (from negative infinity to infinity).

    The problem also gives this morsel of ambiguity:

    "Hint: think before integrating. this is easy"

    Now my initial guess because of the antisymmetry of the function is that it equals zero. Although the problem doesn't ask for a proof of any way shape or form however, I was baffled how I would argue that I reasoned it equaled zero if I was called upon in class.

    So I'm wondering whether my assumption is correct as well as maybe a brief explanation. No proof needed.
  2. jcsd
  3. Apr 3, 2007 #2

    Gib Z

    User Avatar
    Homework Helper

    Exactly correct.
  4. Apr 3, 2007 #3
    A pedant might ask for proof that you can use the antisymmetry of the integral in this way when the limits are +- infinity.

    But I guess that's why we have mathematicians.
  5. Apr 3, 2007 #4

    Gib Z

    User Avatar
    Homework Helper

    [tex]\lim_{a\to\infty}( \int^a_{-a} \sin^3 x dx )

    = \lim_{a\to\infty} (\int^a_0 \sin^3 x dx + \int^0_{-a} \sin^3 x dx)

    =\lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(-a))[/tex], Where dF(x)/dx=sin^3 x.

    Since the derivative of any odd function is an even function, F(-a)=F(a)

    [tex]\lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(-a)) = \lim_{a\to\infty} (F(a) - F(0)) - (F(0) + F(a))

    =\lim_{a\to\infty} (0)

    = 0[/tex].
  6. Apr 3, 2007 #5


    User Avatar
    Staff Emeritus
    Science Advisor

    What Gib Z gives is the "Cauchy Principal Value" of the integral. Of course, the limit is 0 because sin(x) is an odd function. Evaluating its integral at a and -a will give the same thing.

    Strictly speaking [itex]\int_{-\infty}^\infty f(x)dx[/itex] is
    [tex]\lim_{a\rightarrow -\infty}\int_a^0 f(x)dx+ \lim_{b\rightarrow \infty} \int_0^b f(x) dx[/tex]
    where the two limits are taken independently. Using that definition,
    [tex]\int_{-\infty}^\infty sin^3(x) dx[/tex]
    does not exist.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: A sinusoid integrated from -infinity to infinity