Is it possible to integrate the following indefinite integral analytically? $$\int \left(\nabla f\cdot\frac{dx(t)}{dt}\right) \frac{d z(t)}{dt} dt$$
Where $f(x_1(t),x_2(t)):(x_1(t),x_2(t))\to \mathbb{R}^+$, and $x_1(t),x_2(t),z(t)$ are real continuous functions.
Notes
Initially I thought \begin{align} &\int \left(\nabla f\cdot\frac{dx(t)}{dt}\right) \frac{d z(t)}{dt} dt\\ &=\int\left(\nabla f\cdot\frac{dx(t)}{dt}\right) dz(t)\\ &=\left(\nabla f\cdot\frac{dx(t)}{dt}\right) z(t)\\ \end{align} Based on the separation of variables used to solve separable ODEs/PDEs. Obviously $\left(\nabla f\cdot\frac{dx(t)}{dt}\right) z(t)$ is incorrect as \begin{align} \left(\nabla f\cdot\frac{dx(t)}{dt}\right) z(t) \not= \left(\nabla f\cdot\frac{dx(t)}{dt}\right) \frac{d z(t)}{dt}.\\ \end{align}
Is there any technique (maybe from calculus of variations?) that will allow me to integrate this integral?
Context
- If you need clarification feel free to ask.
- This is part of a much larger problem in trying to find a Lypunov function for a systems of ODEs (No $f$ is not a Lypunov function). I have reduced the problem of finding the Lypunov function to finding the integrals similar to the question above.