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Homework Help: Solve integral

  1. Apr 11, 2010 #1
    1. The problem statement, all variables and given/known data
    Solve integral

    [tex]\int \frac{(1+\Phi)+\Phi e^{-x}}{(1+\Phi)-\Phi e^{-x}}dx[/tex]

    where [tex]\Phi=const[/tex]




    2. Relevant equations



    3. The attempt at a solution

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

    [tex](1+\Phi)\int\frac{dx}{(1+\Phi)-\Phi e^{-x}}[/tex]

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

    [tex]\Phi e^{-x}dx=dt[/tex]

    [tex]\frac{1}{1+\Phi-t}=\frac{A}{1+\Phi-t}+\frac{B}{t}[/tex]

    I got

    [tex]A=B=\frac{1}{1+\Phi}[/tex]

    So I got if I don't write constant

    [tex](1+\Phi)\int\frac{dx}{(1+\Phi)-\Phi e^{-x}}=ln[\frac{(1+\Phi)-\Phi e^{-x}}{\Phi e^{-x}}][/tex]

    For second integral I got without constant

    [tex]\int\frac{\Phi e^{-x}}{(1+\Phi)-\Phi e^{-x}}dx=ln[(1+\Phi)-\Phi e^{-x}][/tex]


    So


    [tex]\int \frac{(1+\Phi)+\Phi e^{-x}}{(1+\Phi)-\Phi e^{-x}}dx=2ln[\frac{1+\Phi-\Phi e^{-x}}{\Phi e^{-x}}]+C[/tex]

    Is this solution correct? Thanks for your answer!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Apr 11, 2010 #2

    rl.bhat

    User Avatar
    Homework Helper

    Rewrite the given problem as
    1 + [2φ*e^-x/(1 + φ - φe^-x).......(1)
    If t = 1 + φ - φe^-x
    dt = .........?
    Substitute these values in eq. 1 and find integration.
     
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