I have the process,
$$\gamma_t = \exp\{\int_0^t \sigma_s dW_s-\frac{1}{2} \int_0^t \sigma_s^2 ds\}$$
And I wish to apply Ito's formula. My course provides the proof:
Denote the exponent in $\gamma_t$ by $X_t$. Then by Ito's formula,
$$d\gamma_t = \gamma_t dX_t + \frac{1}{2}\gamma_t(dX_t)^2 = \gamma_t (\sigma_tdW_t - \frac{1}{2}\sigma^2 dt) + \frac{1}{2} \gamma_t \sigma^2 dt = \gamma_t\sigma_tdW_t$$
I am unsure as to why we have an extra $\sigma$ in the second term since just, $(dX_t)^2=dt$ in the second term.
Apologies if I have made some simple mistake