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: 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?

    Thanks
     
  2. jcsd
  3. Feb 21, 2005 #2

    NateTG

    User Avatar
    Science Advisor
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook