Best way to integrate this ?

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  1. member910132's Avatar
    1 30-05-2012  10:27
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    Best way to integrate this ?
    \displaystyle \int coth x \ dx

    I think the best way is to write:
    \displaystyle \int \frac{coshx}{sinhx} \ dx

    But I would also like to be able to do it like this:

    \displaystyle \int \frac{1}{tanhx} \ dx

    So I let u = tanh x and get stuck at:

    \displaystyle \int \frac{1}{u-u^3} \ dx

    How should I proceed from here ? is it a case of splitting the bottom into u(1-u^2) and using partial fractions ? Can anyone show me how to do that step ?

    Thnx
    Last edited by member910132; 30-05-2012 at 11:19.
  2. notnek's Avatar
    2 30-05-2012  10:35
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    • Location: Bangkok, Thailand
    Re: Best way to integrate this ?
    \displaystyle u-u^3 = u(1-u^2) = u(1+u)(1-u)

    So you can use partial fractions:

    \displaystyle \frac{1}{u(1+u)(1-u)} = \frac{A}{u}+\frac{B}{1+u} + \frac{C}{1-u}

    Can you carry on from here?
  3. member910132's Avatar
    3 30-05-2012  11:18
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    Re: Best way to integrate this ?
    (Original post by notnek)
    \displaystyle u-u^3 = u(1-u^2) = u(1+u)(1-u)

    So you can use partial fractions:

    \displaystyle \frac{1}{u(1+u)(1-u)} = \frac{A}{u}+\frac{B}{1+u} + \frac{C}{1-u}

    Can you carry on from here?
    PRSOM
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