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Homework Help: Help with this integral

  1. Jun 27, 2010 #1
    1. The problem statement, all variables and given/known data

    integral of (1+e^x)/(1-e^x) dx

    2. Relevant equations



    3. The attempt at a solution
    The TA said to make u = e^x
    So, du = e^x dx. dx = du/e^x.
    Since e^x = u

    The integral now is (1+u)/(1-u)u
    I am confused as to what to do after distribute the u in the bottom.
     
  2. jcsd
  3. Jun 27, 2010 #2

    vela

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    Use partial fractions.
     
  4. Jun 27, 2010 #3
    alright so i got A = 1 and B = 2 so now i have the integral of 1/u + 2/(1-u). Which i end up getting ln |u| - 2ln|1-u| + c. Now replacing the u for e^x i get ln e^x - 2 ln(1-e^X). Which is x-2 ln (1-e^x) + c. Is that the right answer?
     
  5. Jun 27, 2010 #4

    vela

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    Here's an alternate method you can use to solve this problem.

    Whenever I see combinations of [itex]1\pm e^x[/itex], I look at what happens if I pull out a factor of [itex]e^{x/2}[/itex] to restore symmetry to the quantities. In this case, the integrand becomes

    [tex]\frac{1+e^x}{1-e^x} = \frac{e^{x/2}(e^{-x/2}+e^{x/2})}{e^{x/2}(e^{-x/2}-e^{x/2})} = \frac{e^{-x/2}+e^{x/2}}{e^{-x/2}-e^{x/2}}[/tex]

    You might notice that the top and bottom can be written in terms of hyperbolic trig functions or that they are derivatives of each other to within a constant factor. In either case, with the appropriate substitution, integrating is straightforward.
     
  6. Jun 27, 2010 #5

    Mark44

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    Check it yourself and see if it's the right answer. If the derivative of your answer equals the integrand, then your answer is correct.
     
  7. Jun 27, 2010 #6

    vela

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    It's easy enough to check. Try differentiating your answer and see if you recover what you started with.
     
  8. Jun 27, 2010 #7
    wow that makes is so much simpler..thanks so much!
     
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