Variance of the Ito integral of W dW

$W_t$ is a standard Brownian motion. The variance of $\displaystyle\int_0^t W_s\,dW_s$ equals $kt^2$ for some constant $k$. Determine $k$.

$k=\tfrac12$. Ito's formula on $W_t^2$ gives $\int_0^tW_s\,dW_s=\tfrac12(W_t^2-t)$, whose variance is $\tfrac14\operatorname{Var}(W_t^2)=\tfrac14\cdot2t^2$. Equivalently, by the Ito isometry the variance is $\int_0^t\mathbb{E}[W_s^2]\,ds=\tfrac{t^2}{2}$.