Finite difference method for conservative form of equations

My question is about how do we discretize the equations in the conservative form using finite difference method.

I'm trying to solve Euler equations in conservative form. $$\frac{\partial u}{\partial t}+\frac{\partial F(u)}{\partial x}=0$$

I'm confused with the following two approaches. $$\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+\Delta x/2}^n-F_{i-\Delta x/2}^n}{\Delta x}=0$$

and the other one is $$\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+1}^n-F_{i-1}^n}{2\Delta x}=0$$

What is the right way of discretization and how are they different. It would be really helpful if you could point me out to some literature. I have been looking all over.

After a bit of research, I figured out that Case: 1 $$\frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}+\frac{F_{i+\Delta x/2}^n-F_{i-\Delta x/2}^n}{\Delta x}=0$$ can be used while incorporating TVD or WENO. Where as the second case is pretty straight forward for generic discretizations.