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Math Help - integrate (f*g')/g

  1. #1
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    integrate (f*g')/g

    Hi all
    Are there common ways to manipulate the integral


    \int\frac{fg'}{g}dx

    so that it can be evaluated? would one need to know the exact functions in question or can this be answered generally?
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  2. #2
    Moo
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    Hello,

    Nope it can't be solved without further information. It's not a known form
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  3. #3
    MHF Contributor chisigma's Avatar
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    Calling x the independent variable is...

    \frac{g^{'}(x)}{g(x)}= \frac{d}{dx} \ln g(x) (1)

    ... so that integrating by parts we have the identity...

    \int f(x)\cdot \frac{g^{'}(x)}{g(x)}\cdot dx = f(x)\cdot \ln g(x) - \int f^{'}(x)\cdot \ln g(x)\cdot dx (2)

    Honestly I don't know if (2) can be of some utility for You ...

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor chisigma's Avatar
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    An interesting possibility is when is f(x)= g(x) + \chi with  \chi an arbitrary constant, so that is f^{'} (x) = g^{'} (x). In such a case is...

    \int f(x)\cdot \frac {g^{'} (x)}{g(x)}\cdot dx= 2\cdot g(x)\cdot \ln g(x) + (\chi -1)\cdot g(x) + c

    Kind regards

    \chi \sigma
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