Computing the Ito integral of w against itself

For a standard Brownian motion $w$, evaluate $\int_0^T w(t)\,dw(t)$ in closed form and describe the properties of the resulting random variable.

$\tfrac12\big(w(T)^2-T\big)$. Ito's lemma on $w^2$ gives $d(w^2)=2w\,dw+dt$; the extra $-T$ is the Ito correction that ordinary calculus misses. The result has mean $0$ and variance $T^2/2$, is a martingale in $T$, and is distributed as $\tfrac T2(\chi^2_1-1)$, so it is skewed and bounded below by $-T/2$.