The paper suggests a way of stochastic integration of random integrands with respect to fractional Brownian motion with the Hurst parameter H> 1/2. The integral is defined initially on the processes that are "piecewise" predictable on a short horizon. Then the integral is extended on a wide class of square integrable adapted random processes. This class is described via a mild restriction on the growth rate of the conditional mean square error for the forecast on an arbitrarily short horizon given current observations; differentiability of Hölder property of any kind or degree is not required for the integrand. The suggested integration can be interpreted as foresighted integration for integrands featuring corresponding restrictions on the forecasting error. This integration is based on Itô's integration and does not involve Malliavin calculus or Wick products. In addition, it is shown that these stochastic integrals depend continuously on H at H=1/2+0.