numerical integration of a function satisfying a ode

Question

I need to numerically approximate an integral of the form $$\int_0^\tau f(X_t)\:{\rm d}t,\tag1$$ where $(X_t)_{t\ge0}$ is the solution of a SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag2.$$ For simplicity, let's assume that $\sigma=0$ and hence $(2)$ is an ODE. In that case, $${\rm d}f(X_t)=\left[b(X_t)\cdot\nabla f(X_t)\right]{\rm d}t\tag3$$ and hence we can write $(1)$ as $$\tau f(X_\tau)+\int_0^\tau\tau b(X_t)\cdot\nabla f(X_t)\:{\rm d}t\tag4.$$ The question is: Does $(4)$ allow to come up with a "smarter" numerical integration scheme for $(1)$? A general scheme, like $$(1)\approx\frac1\tau\sum_{i=0}^{k-1}(t_{i+1}-t_i)f(X_{t_i})\tag5$$ has no knowledge about $t\mapsto f(X_t)$ incorporated. So, I could imagine that there are special schemes in the present situation.