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How to integrate this?

  1. Oct 23, 2014 #1
    1. The problem statement, all variables and given/known data
    How to integrate dz/(z^2+2z/x)?

    2. Relevant equations
    None.

    3. The attempt at a solution
    I did partial fractions but it doesn't seem to work.
    A/z+B/(z+2/x)=1
     
  2. jcsd
  3. Oct 23, 2014 #2

    ShayanJ

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    Gold Member

    Try completing the square!
     
  4. Oct 24, 2014 #3
    The thing to do for partial fractions would be to solve:
    1/(z^2+2z/x) = 1 / z(z+2/x) = A/z+B/(z+2/x)
    for A and B, and then it can be integrated to logs.

    Or, you can complete the square and you're left with an integral that yields a hyperbolic trig solution which can be expressed in terms of logs:
    http://en.wikipedia.org/wiki/Hyperbolic_function

    The interesting thing about this problem is that it has simple poles at z=0 and -2/x.
    So you get log absolute values when integrating 1/z at negative z...,
    and there are corresponding domain restrictions on the validity of the hyperbolic trig solution.
    When you take care, you can arrive at the two methods yielding the same solution over their common domain of validity: z=(-2/x,0) (assuming x>0).
     
  5. Oct 24, 2014 #4

    vela

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    Staff Emeritus
    Science Advisor
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    Education Advisor

    That would be because the righthand side shouldn't be equal to 1.
     
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