that x is the variable that you perform the integral over. Variable of integration.
It comes from the defintion of integral, that you sum over the function value multiplying it with a very very small x-intervall (length multiplied with height). Riemann summary. If we make the intervall infinitesimal (dx), the we have to do, in practice, an infinite amount of summations over these "function value" * "length of intervall"
I'm reviewing Meirovitch's "Methods of Analytical Dynamics," and I don't understand the commutation of the derivative from r to dr:
$$
\mathbf{F} \cdot d\mathbf{r} = m \ddot{\mathbf{r}} \cdot d\mathbf{r} = m\mathbf{\dot{r}} \cdot d\mathbf{\dot{r}}
$$