I am currently trying to use the Ito formula to evaluate the expression
$\int_0^t(W_1^2(u)W_2^2(u)+u(W_1^2(u)+W_2^2(u)))du+2\int_0^tuW_1(u)W_2^2(u)dW_1(u)+2\int_0^tuW_1^2(u)W_2(u)dW_2(u)$
where $W_1$ and $W_2$ are independent Brownian motions.
I so far have tried to applied the multidimensional Ito formula, where I set $f=tW_1^2W_2^2$, which gave me
$df=\int_0^tu(W_1^2(u)+W_2^2(u))du+2\int_0^tuW_1(u)W_2^2(u)dW_1(u)+2\int_0^tuW_1^2(u)W_2(u)dW_2(u)$, which is nearly right, though I'm still missing $\int_0^t(W_1^2(u)W_2^2(u))du$. Have I simply picked the wrong function for the Ito formula, or am I missing something else here?