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Integral variable substitution for removing singularity

  1. Jan 10, 2012 #1
    The problem statement, all variables and given/known data
    Hi! In an assignment I have reached an integral that has the form:
    [itex]\int\frac{A+Bx+Cx^2}{Dx+Ex^2}[/itex]
    where A-E are constants, the integration variable is x and the limits are 0 to 1. I'm supposed to remove the singularity at x=0 by substitution.

    A-E have values but they're long and complicated and I hope they're not necessary to solve the problem. And sadly, no - I can't go backwards to complete squares or anything...


    The attempt at a solution
    This might be an easy question, but I really don't know what to substitute x for. I've tried squares, roots, inverted squares and roots, ln and exponential functions... but they all end up with the same singularity at 0. I don't know how to get around this. Is there any good way to figure out how to substitute in order to "remove" a singularity, in general?

    Help and hints would be very much appreciated! Thank you!

    /Jennifer
     
  2. jcsd
  3. Jan 10, 2012 #2

    SammyS

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    Try the following. Not sure how well it works for the whole problem.

    Let u = 1/x . Of course that's equivalent to x = eu .
     
  4. Jan 10, 2012 #3
    Suggestion: Divide using long, polynomial division and rewrite the integral :)
     
  5. Jan 10, 2012 #4

    SammyS

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    That still leaves a singularity at x=0.
     
  6. Jan 10, 2012 #5
    Ahh. Thanks for your suggestions, they made me re-track my steps. And yep, I left out the part that the whole fraction is under a square root as well :tongue: That changes things a bit!

    Substituting x with u2 seems to do the trick now! Phew :redface:
     
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