$\int \frac{f\left(x\right)\text{'}-f\text{'}\left(x\right)g\left(x\right)}{f\left(x\right).g\left(x\right)}\left(\log g\left(x\right)-\log f\left(x\right)\right)dx=$

$\int \frac{f\left(x\right)\text{'}-f\text{'}\left(x\right)g\left(x\right)}{f\left(x\right).g\left(x\right)}\left(\log g\left(x\right)-\log f\left(x\right)\right)dx$

$$\begin{gathered} =\int \frac{f\left(x\right).g\text{'}\left(x\right)-g\text{'}\left(x\right)f\text{'}\left(x\right)}{f{\left(x\right)}^2}\left(\frac{g\left(x\right)}{f\left(x\right)}\right)\log \left(\frac{g\left(x\right)}{f\left(x\right)}\right)dx \\ =\int d\left(\frac{g\left(x\right)}{f\left(x\right)}\right)\frac{1}{g\left(x\right)}\times \log \left(\frac{g\left(x\right)}{f\left(x\right)}\right) \\ =\int \frac{\log t}{t}.dt where \frac{g\left(x\right)}{f\left(x\right)} \\ =\frac{{\left(\log t\right)}^2}{2}+C \\ =\frac{1}{2}\log {\left(\frac{g\left(x\right)}{f\left(x\right)}\right)}^2+C \end{gathered}$$