Consider a function $F=F(q)$ which is a symmetric, positive-definite matrix form of dimensions $n\times n$, where $q \in \mathbb{R}^n$ and $q=q(t)$.
When applying the midpoint integration rule on $F$, it is straightforward that $$ \int_0^h F(q)\,\mathrm{d}t \simeq h F\left(\frac{q_\ell + q_r}{2}\right) $$ where subindices $\ell$ and $r$ stand for "left" and "right" values respectively.
However, what would be the correct way to establish the following integral using the midpoint rule?
$$ \int_0^h \dot q^\intercal F(q)\, \dot q $$
EDIT
I want to consider a centered finite difference for $\dot q$, such that
$$\dot q = \frac{d q}{dt} = \frac{q_r - q_\ell}{h},$$
where $h$ is the time step.