It is well known that the integral $$\int_0^t W_s ds,$$ where $(W_s)_s$ is a Brownian motion, can be derived using Ito's Lemma. More precisely, Ito's lemma on $d(tW_t)$ implies that $$d(tW_t) = tdW_t + W_t dt.$$ Therefore, $$\int_0^t W_s ds = tW_t - \int_0^t sdW_s.$$ Its mean and variance can be obtained from this expression. This leads to my question below.
Question: Given a positive integer $n,$ what is the mean and variance $$\int_0^t (W_s)^n ds?$$
Calculation above is for $n=1.$