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Integral of sinhx.lncosh^2(x).dx

  1. Dec 8, 2011 #1

    sharks

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    The problem statement, all variables and given/known data
    Integral of sinhx.lncosh^2(x).dx

    The attempt at a solution
    I used integration by parts.
    Let U = lncoshx^2(x)
    Let dV = sinhx

    So, dU =sech^2(x)/(2coshx.sinhx) = 1/(2cosh^3(x).sinhx) [i hope this line is correct?]
    And, V = coshx

    When using the formula for integration by parts, the integral of V.dU proves to be difficult. I think i should use substitution to simply it somehow, but not sure how.
     
  2. jcsd
  3. Dec 8, 2011 #2

    dextercioby

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    Substitute [itex] \cosh x = t [/itex].
     
  4. Dec 8, 2011 #3

    sharks

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    So, i used the substitution suggested above into V.dU, and i get:

    Integral of V.dU = Integral of 1/[2(t^4 - t^2)] w.r.t.t

    I could use this equivalent (re-arranged) expression: Integral of 1/[2(t^4 - t^2)] w.r.t.t = Integral of 1/[(2t^2).(t^2 - 1)] w.r.t.t

    But then i would probably have to integrate this by parts again. I guess i'm stuck once more.
     
    Last edited: Dec 8, 2011
  5. Dec 8, 2011 #4

    SammyS

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    That line is incorrect.

    [itex]\displaystyle \frac{d}{dx}\ln(\cosh^2(x))= \text{sech}^2(x)\cdot(2\cosh(x)\sinh(x))[/itex]
    [itex]\displaystyle =2\,\text{sech}(x) \sinh(x)[/itex]​
    Of course that could have been arrived at more easily by noticing that [itex]\displaystyle \ln(\cosh^2(x))=2\ln(\cosh(x))\,.[/itex]
    Your integration by pars should work fine, except,
    dv should be: [itex]\displaystyle dv=\sinh(x)\,dx\,.[/itex]

    This should result in [itex]\displaystyle\int v\,du=2\int\sinh(x)\,dx\,.[/itex]
     
  6. Dec 9, 2011 #5

    sharks

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    I finally got it. Thanks SammyS for the hint about 2ln(cosh x). I used that and it all got much simpler, like you said.
     
  7. Dec 9, 2011 #6

    dextercioby

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    I mean without part integration at the beginning. First substitute, then the new integral should be ~ int ln t dt. Now you can use part integration to compute the antiderivative of the logarithm (which is also tabluated).
     
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