1. integrate (f*g')/g

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?

2. Hello,

Nope it can't be solved without further information. It's not a known form

3. 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

$\displaystyle \chi$ $\displaystyle \sigma$

4. 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

$\displaystyle \chi$ $\displaystyle \sigma$