CONTEXT (can skip):
My textbook looks at two things -
1) Ito integrals for deterministic functions—i.e. $\int f(t) \,dW_t$. We are able to say that they are normally distributed, with a mean of 0 and a variance of $\int f^2(t) \,dt$.
2) Ito integrals for stochastic functions—i.e. $\int f(W_t, t) \,dW_t$. We aren't able to find their distribution in general; but we can conclude that they have a mean of 0, and ito isometry can be used to get a neat expression for their variance.
Is it meaningful to look at something of the form $\int f(W_t, t) \,dt$?
MY ANSWER (possibly wrong):
I feel the answer is yes. We won't be able to find the distribution of $\int f(W_t, t) \,dt$ in general, but we can say that it will have a mean of $\int E[f(W_t, t)] \,dt$. I also think that the second order raw moment $E[(\int f(W_t, t) \,dt)^2]$ will be given by $\int \int E[f(W_t, t) f(W_s, s)] \,dtds$, which can then be used to find the variance.