Consider $\int_{t}^{t + \Delta t} a W_s dW_s $, where $a$ is constant.
How do I get the following approximation of this integral: $$ a \Delta W_t + \frac{1}{2} a^2 (\Delta W_t)^2$$
Where does the scond term come from? Why do I need that since the integral evaluates to $a (W_{t + \Delta t} -W_t)$?