Variance of integrated squared wiener process

Question

So I'm trying to figure out the mean and variance of

$X = \int_{0}^{1} W^2(t) dt $

where $W$ is the Wiener process.

The mean I've worked out easily to be $\frac{\sigma^2}{2}$ but I'm having trouble with the variance. I am unable to find $E(X^2)$ in order to calculate $Var(X) = E(X^2) - (E(x))^2$ but something tells me this isn't the right way to approach this problem. Any nudge in the right direction would be appreciated.

Note: This is practice for an upcoming exam, not homework.

Answer 1

Hint: $$E(X^2)=2\int_0^1\int_0^tE(W(t)^2W(s)^2)\mathrm ds\mathrm dt$$ Can you compute $E(W(t)^2W(s)^2)$ for $s\lt t$?