It is well known that $$ \int \frac{f'(x)dx}{f(x)}= \int d \log f(x)=\log f(x) + C $$
In my work I came across the following case: $$ \int \frac{(f'(x))^2dx}{f(x)} $$
I wonder if any interesting approach exists in this situation
It is well known that $$ \int \frac{f'(x)dx}{f(x)}= \int d \log f(x)=\log f(x) + C $$
In my work I came across the following case: $$ \int \frac{(f'(x))^2dx}{f(x)} $$
I wonder if any interesting approach exists in this situation
Integrating by parts gives,
$$ \int \frac{f'(x)f'(x)dx}{f(x)} = f'(x)\ln(f(x)) - \int f''(x)\ln(f(x))dx \,.$$