Hi all
Are there common ways to manipulate the integral
$\displaystyle \int\frac{fg'}{g}dx$
so that it can be evaluated? would one need to know the exact functions in question or can this be answered generally?
Calling $\displaystyle x$ the independent variable is...
$\displaystyle \frac{g^{'}(x)}{g(x)}= \frac{d}{dx} \ln g(x)$ (1)
... so that integrating by parts we have the identity...
$\displaystyle \int f(x)\cdot \frac{g^{'}(x)}{g(x)}\cdot dx = f(x)\cdot \ln g(x) - \int f^{'}(x)\cdot \ln g(x)\cdot dx$ (2)
Honestly I don't know if (2) can be of some utility for You ...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
An interesting possibility is when is $\displaystyle f(x)= g(x) + \chi$ with $\displaystyle \chi$ an arbitrary constant, so that is $\displaystyle f^{'} (x) = g^{'} (x)$. In such a case is...
$\displaystyle \int f(x)\cdot \frac {g^{'} (x)}{g(x)}\cdot dx= 2\cdot g(x)\cdot \ln g(x) + (\chi -1)\cdot g(x) + c$
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$