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Weird integral - what's wrong?

  1. May 17, 2015 #1
    Heya...

    I just stumbled upon this integral while doing ODEs...

    ∫x*(x-2)-2dx

    I used integration by parts

    u=x
    du/dx=1
    dv/dx=(x-2)-2
    v=-(x-2)-1

    Thus

    ∫x*(x-2)-2dx=-x(x-2)-1-∫1*-(x-2)-1dx=-x*(x-2)-1+ln(x-2)+C

    Now... I don't see anything wrong with this...

    But every single online integration calculator as well as the ODE solution solves this integral by doing 1/(x-2)+2/(x-2)2 and they all get

    -2*(x-2)-1+ln(x-2)+C

    as a result...

    What am I doing wrong in my integration by parts? What am I missing? Oo
     
  2. jcsd
  3. May 17, 2015 #2

    mathman

    User Avatar
    Science Advisor
    Gold Member

    There is nothing wrong. The two results are equivalent.

    [itex]\frac{x}{x-2}=\frac{x-2+2}{x-2}=1+\frac{2}{x-2}[/itex].

    The difference is only in the constant of integration.
     
  4. May 17, 2015 #3
    Thank you!
    The maths department screwed up then because the question asked to verify that C=2 but with the above integration it equals 3... So they omitted one result... >.<
     
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