I have
\begin{align} x' &= y\\ y' &= -f(x)y -xg(y)\\ \end{align}
$f,g$ are functions, especially $f \ge 0$. I could confirm that $(0,0)$ is equilibrium point easily.
I want to confirm that $L_F := \frac{1}{2} y^{2} + \int_0^{x} sg(s) ds$ is Lyapunov function.
And more, Defining $F$ as
$F \left( \begin{array}{c} x\\ y \end{array} \right) = \left( \begin{array}{c} -y\\ -f(x)y -xg(y) \end{array} \right)$,
I want to confirm that $(0,0)$ is asymptotic stability with Lyapunov stability theorem.
How can I confirm them? I have difficulty in differentiating them.
Your Lyapunov function candidate is positive definite. This is the first criterium that you need to show that $L_f$ is a Lyapunov function.
The second criterium is that the derivative along your system of ODEs is negative (semi) definite. (semi definiteness shows stability of the equilibrium point, definiteness shows asymptotic stability).
Taking the derivative of $L_f$ gives: ( note, I assume here you mean $g(x)$ at the right hand side of your ODE and not $g(y)$)
$\dot L_f = y y’ + x’ (d/dx \int s g(s) ds) = -y^2 f - x y g + x y g = -y^2 f\leq 0$
This shows that $L_f$ is a Lyapunov function.
Hence, it guarantees that the equilibrium is stable as the derivative is negative semi definite. For this system you can still show asymptotic stability by using La Salle’s invariant set theorem.