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Periodic function and substitution question!

  1. May 19, 2009 #1
    Let f: R-> R be a continuous function. Let T>0 be such that
    f(x+T)= f(x) for all x.
    We say that f is a periodic function with a period T>0.
    Use an appropriate substitution to prove that for all real numbers a
    [tex]\int^{a+T}_{a}f(x)dx[/tex] = [tex]\int^{T}_{0}f(x)dx[/tex].

    I have no idea how to do this question.
    thanks for helping me!
  2. jcsd
  3. May 19, 2009 #2


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    Gold Member

    Try breaking the integral on the left into two parts:

    [tex]\int_{a}^{a+T} = \int_{a}^{T} + \int_{T}^{T+a}[/tex]

    and in the second part, use the fact that f is periodic.
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