Integral of $\frac{1}{f(x)}(\frac{df}{dx})^{2}$

Question

I'm trying to find a way to simplify or solve an integral involving a function $f(x)$ of $x$.

Here it is:

$\int\frac{1}{f(x)}(\frac{df}{dx})^{2}dx$

I tried integrating by parts a few times, but wasn't sure if I was on the right path.

This integral arose when I was trying to integrate another function by parts:

$\int\frac{d^2f}{dx^2}ln[f(x)]dx$

Anyways, is this even possible to solve?

Answer 1

There is no bivariate function $L(x,y)$ such that $$\mathrm{d}L=\tfrac{(y')^2}{y}\mathrm{d}x\text{.}$$ If there were, then we would have $$L_{,x}+L_{,y}y'=\tfrac{(y')^2}{y}\text{;}$$ but the left side is linear in $y'$ whereas the right side is quadratic. Similar arguments apply for $L(x,y,y')$.

Consequently, your integration procedure must depend on the details of $f$.