1
$\begingroup$

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?

$\endgroup$
0
$\begingroup$

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 $f$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.