I am wondering what is the distribution of:
$$ \int_0^tW_sdW_s $$
Solution: (Thanks to @muaddib)
Applying Ito's Formula to $W_t^2$ gives $d(W_t^2) = 2W_tdW_t +dt$, and so: $$ \int_0^tW_sdW_s= W_t^2 - t $$ Since $W_t^2 \sim ~t Z^2$ where $Z\sim N(0,1) \implies Z^2 \sim \chi^2_1 $ , then:
$W_t^2 \sim \Gamma(\frac{1}{2},2t)$