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How do you integrate dirac delta functions?

  1. Mar 31, 2015 #1
    1. The problem statement, all variables and given/known data
    ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞.

    2. Relevant equations


    3. The attempt at a solution
    Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would be 1/√5.
    But i'm unsure how to go about this one. Wolfram Alpha states the answer is 1/6. Tried to factor the polynomial which results in (x-1)(x+2)(x-5). But that doesn't seem to help either.

    Any guidance would be great thanks!
     
  2. jcsd
  3. Mar 31, 2015 #2

    Orodruin

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    You are on the right track. You now know what points give a contribution (the roots of the polynomial) if they are inside your integration interval.

    Now, do you remember where the factor 1/sqrt5 comes from in the case of x^2-5?
     
  4. Mar 31, 2015 #3
    Are you saying that I should only consider the (x-5) and (x-1) roots? Would that then result in the answer being 1/6? Why would I not consider the negative root? :S
    I know this relation exists:

    ∫dx⋅f(x)⋅δ(ax+b) = 1/|a|⋅f(-a-1b). (sorry for the formatting) (limits ±∞ again)

    Which I understand makes sense for say δ(2x). Not quite sure how √5 is a in δ(x^2-5). But I get the system. Then i'm really confused for the 3 order polynomial above.




    https://www.physicsforums.com/file:///page36image21544 [Broken] https://www.physicsforums.com/file:///page36image21704 [Broken]
     
    Last edited by a moderator: May 7, 2017
  5. Mar 31, 2015 #4

    Ray Vickson

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    [tex] \delta(x^2-5) = \delta((x-\sqrt{5})(x + \sqrt{5})) = \delta(2 \sqrt{5} (x - \sqrt{5}) + \delta(-2 \sqrt{5}(x + \sqrt{5})) [/tex]
     
    Last edited by a moderator: May 7, 2017
  6. Mar 31, 2015 #5
    ...Thank you, i'm an idiot.
     
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