Compute the first and second moments of the time integral \(\int_{\tau}^{T} S_{t} d t\) for \(\tau \in[0, T)\), where \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)is the geometric Brownian motion \(S_{t}:=S_{0} \mathrm{e}^{\sigma B_{t}+r t-\sigma^{2} t / 2}, t \geqslant 0\).