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Improper integrals

  1. Feb 21, 2005 #1
    If [tex]\int _{-\infty} ^{\infty}f(x)\: dx[/tex] is convergent and [tex]a[/tex] and [tex]b[/tex] are real numbers, show that

    [tex]\int _{-\infty} ^a f(x)\: dx + \int _a ^{\infty}f(x)\: dx = \int _{-\infty} ^b f(x)\: dx + \int _b ^{\infty}f(x)\: dx[/tex]

    I'm clueless on how to show it other than by drawing what is stated: a generic finite integral being split into two finite pieces for each arbitrary point. Is there any other way to approach this problem?

  2. jcsd
  3. Feb 21, 2005 #2


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    Perhaps you could use something like:
    [tex]\int _{-\infty} ^a f(x)\: dx + \int _a ^{\infty}f(x)\: dx = \int _{-\infty} ^a f(x)\: dx + \int _{a} ^b f(x)\: dx +\int _b ^{\infty}f(x)\: dx = \int _{-\infty} ^b f(x)\: dx + \int _b ^{\infty}f(x)\: dx[/tex]

    But that's really just an algebraic representation of what you're suggesting.
  4. Feb 21, 2005 #3
    That sounds about right. I mean, I can't see anything else that could be done.
    Last edited: Feb 21, 2005
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