In some financial application, we are given that
$$X(t)=B(t)\times[{X(0)+\int_{0}^{t}\lambda(s)dW(s)}]$$
and need to calculate the $dX(t)$ using Ito's product rule. Assuming $X(0)$ is given, then $d[X(0)+\int_{0}^{t}\lambda(s)dW(s)]$ is simply $\lambda(t)dW(t)$ and if we know that $dB(t)=\alpha(t)B(t)dt$ then we have
$$dX(t)=\alpha(t)B(t)dt\times[{X(0)+\int_{0}^{t}\lambda(s)dW(s)}] + B(t)\times \alpha (t)B(t)dt$$
(no third term as $dtdW(t)$ cancel out.)
Is there a way to simplify the first term $\alpha(t)B(t)dt\times[{X(0)+\int_{0}^{t}\lambda(s)dW(s)}]$, i.e. how do you deal with $(...)dt\times \int_{0}^{t}...dW(s)$?