Solution
= Solution
{c}
For the time integral $I_t=\int_0^tW_sds$, <Fubini's theorem> and $\mathbb EW_s=0$ give $\mathbb EI_t=0$. Brownian covariance is $\mathbb E[W_uW_v]=\min(u,v)$, so
$$
\operatorname{Var}(I_t)=\int_0^t\int_0^t\min(u,v)\,du\,dv
=2\int_0^t\int_0^v u\,du\,dv=\int_0^t v^2dv.
$$
Consequently
$$
\boxed{\mathbb EI_t=0,\qquad\operatorname{Var}(I_t)=t^3/3.}
$$