Monotonically controlled integrals

The monotonically controlled integral defined by Bendov\'a and Mal\'y, which is equivalent to the Denjoy-Perron integral, admits a natural parameter $\alpha>0$ thereby leading to the whole scale of integrals called $\alpha$-monotonically controlled integrals. While the power of these integrals is easily seen to increase with increasing $\alpha$, our main results show that their exact dependence on $\alpha$ is rather curious. For $\alpha<1$ they do not even contain the Lebesgue integral, for $1\le\alpha\le 2$ they coincide with the Denjoy-Perron integral, and for $\alpha>2$ they are mutually different and not even contained in the Denjoy-Khintchine integral.