I have:

$\displaystyle f=\frac{bt}{m}L-\frac{tx^2}{2}k'(t)x^2e^{\frac{bt}{m}}$

L is a lagrangian so I can be confident that it can be integrated with respect to time.

I want to show that this quantity is the total derivative of some function F. I.e. a function F exists such that:

$\displaystyle \frac {dF}{dt}=f$

1. How would I make such an argument?

2. Can I make an argument without actually fnding F?

3. What features would/could a function f have if this were not true?