It is well known that for sufficiently smooth function $f(t)$, error bounds for midpoint are $$ \int_{t_i}^{t_{i+1}} f(s)ds=hf(t_{i+1/2})+\frac{h^3}{24}\frac{d^2 f(s)}{ds^2}|_{s=t_i} +O(h^5). $$ where $h$ is a constant step-size.
but there are many functions which can be integral-able, but $\frac{d^2 f(s)}{ds^2}$ is not bounded. (e.g. $\frac{1}{\sqrt {t}}$ when we consider interval including $0$).
In the case, can't we use midpoint-method?
if not, how to approximate singular function and what is error bound for that method?