Right now I dont know how to find the expectation and the variance for the following stochastic integral: $$\int_{0}^t B(s) dW(s)$$ where $B(t)$ and $W(t)$ are correlated standard Brownian Motions with $d\langle B, W \rangle_t = \gamma dt$.

Usually, when $B(t)$ and $W(t)$ are independent, we know $$\mathbb{E}\left(\int_{0}^t B(s) dW(s)\right) = 0$$ and by Ito's isometry, $$Var\left(\int_{0}^t B(s) dW(s)\right) = \mathbb{E}\left(\int_{0}^t B(s) dW(s)\right)^2 = \int_{0}^t \mathbb{E}B(s)^2 ds.$$

Does these results still apply?