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Integral - Algebraic Manipulation?

  1. May 21, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]\int \frac{x}{x+d}dx[/tex]

    3. The attempt at a solution

    I tried parts and that turned out horribly so I fired up maple. First move it makes is 'Rewrite Rule'. I've never heard of a rewrite rule but I supposed it could be some algebraic massaging. I ended up looking at it this way:

    [tex]\frac{x}{x+d} = 1-1+\frac{x}{x+d} = \frac{x+d}{x+d}-\frac{x+d}{x+d}+\frac{x}{x+d} = \frac{x+d}{x+d}+\frac{x-d-x}{x+d} = 1-\frac{d}{x+d}[/tex]

    Which then is easily integrable. My question is, is there a snappier way to think of this/do it?
  2. jcsd
  3. May 21, 2008 #2


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    Homework Helper

    Just divide x by x+d and you'll get the term on the rightmost of your equalities.
  4. May 21, 2008 #3
    As in polynomial long division?

    edit: Or is there some way that is easy to see what will happen? Basically it isn't straight up intuitive for me to realize that x/x+d is 1-d/x+d, I'm trying to find a quick method to deal with it.
    Last edited: May 21, 2008
  5. May 21, 2008 #4

    Break it and integrate.
  6. May 21, 2008 #5
    That's much shorter than what I did, thanks.
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