What is the right approach to take and find the moments of the following: $\mathcal{Z}_t=\int\mathcal{W}_t^k\,d\mathcal{W}_t=?$ $\mathcal{W}_t \sim \mathcal{N}(0, t),\ k=2,3...$ $\operatorname{E}(\mathcal{Z}_t)=?$ $\operatorname{Var}(\mathcal{Z}_t)=?$
I know that $\mathcal{Y}_t=\int\mathcal{W}_t\,d\mathcal{W}_t=\frac{\mathcal{W}_t^2-t}{2}$ and $\operatorname{E}(\mathcal{Y}_t)=0$. One can derive it from Ito's lemma and the fact that $\operatorname{Var}(\mathcal{W}_t)=t=\operatorname{E}(\mathcal{W}_t^2)$. Is there another way to prove it?
Is there also some generic rules to deal with $\int t\,d\mathcal{W}_t$ and $\int \mathcal{W}_t\,dt$?
Thanks!