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A tip for integral please

  1. Nov 20, 2006 #1
    Can anybody give me a hint how to solve, if it is possible at all?
    [tex] \int \frac{dv}{-g-kv\sqrt{v^{2}+u^{2}}} [/tex]
     
    Last edited: Nov 20, 2006
  2. jcsd
  3. Nov 21, 2006 #2

    dextercioby

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    I don't think it can be solved in terms of elementary functions.

    Daniel.
     
  4. Nov 22, 2006 #3

    benorin

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    I'll say, and I quote some computer when I say,

    so good job xAxis, you've done quite :bugeye: , no rather :approve: , wait! certianly this is :rofl: . Yes, that's it, I'm quite :rofl: with this integral/error message combo.

    --Ben
     
  5. Nov 22, 2006 #4

    dextercioby

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    It can be solved exactly in terms of elementary functions.

    Daniel.
     
  6. Nov 22, 2006 #5

    dextercioby

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    [tex] I=\int \frac{dx}{-g-kx\sqrt{x^{2}+p^{2}}} =-\frac{1}{k}\int \frac{dx}{\frac{g}{k}+x\sqrt{x^{2}+p^{2}}} [/tex]

    Now make the substitution

    [tex] x=p\sinh t [/tex]

    [tex] I= -\frac{p}{k}\int \frac{\cosh t \ dt}{\frac{g}{k}+p^{2}\sinh t\cosh t}= -\frac{p}{k}\int \frac{\cosh t \ dt}{\frac{g}{k}+\frac{p^{2}}{2}\sinh 2t} [/tex],
     
  7. Nov 22, 2006 #6

    dextercioby

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    [tex] I= -\frac{p}{k}\int \frac{e^{t} +e^{-t}}{\frac{2g}{k}+\frac{p^{2}}{2}\left(e^{2t}-e^{-2t}\right)} \ dt = -\frac{2}{kp}\left(\int \frac{e^t}{\frac{4g}{p^{2}k}+e^{2t}-e^{-2t}} \ dt + \int \frac{e^{-t}}{\frac{4g}{p^{2}k}+e^{2t}-e^{-2t}} \ dt \right) [/tex],

    [tex] I=-\frac{2}{kp}\left(\int \frac{e^{3t}}{e^{4t}+\frac{4g}{p^{2}k}e^{2t}-1} \ dt +\int \frac{e^{t}}{e^{4t}+\frac{4g}{p^{2}k}e^{2t}-1} \ dt \right) [/tex]

    The 2 remaining integrals can be computed exactly.

    Daniel.
     
    Last edited: Nov 22, 2006
  8. Nov 22, 2006 #7

    dextercioby

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    For example:

    See the attached file.

    Daniel.
     

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