Composite midpoint rule error

For $f\in C^2[a,b]$, the composite midpoint rule with mesh width $h$ has local error $O(h^3)$ and global error $O(h^2)$.

If $f(x)=x^{-1/2}g(x)$ with $g$ smooth near zero, the substitution $x=s^2$ givesThe transformed integrand is smooth, so an ordinary second-order midpoint rule recovers its usual convergence rate.