I need to show that
$$ \int_0^t B_s^3 dB_s = \frac{1}{4}B_t^4 - \frac{3}{2} \int_0^t B_s^2 \, ,$$
by only using the following approximations:
$$ \lim_{n \to \infty} \sum_{i=1}^{n} f_{t_{i-1}} \Delta B_{t_i} = \int_0^t fdB_s$$
$$ \lim_{n \to \infty} \sum_{i=1}^{n} \Delta(B_{t_i}^k) = \Delta(B_{t}^k)$$
$$ \lim_{n \to \infty} \sum_{i=1}^{n} f_{t_{i-1}} \Delta s_{i} = \int_0^t f ds$$
and the properties of stochastic integrals.
I guess I need to start by expanding $\Delta(B_{t_i}^4) =B_{t_i}^4 - B_{t_{i-1}}^4 $ and adding and subtracting terms until I get the desired result but I haven't been able to arrive at it yet.