How to prove Ito integral with respect to Iterated Ito Integral

Question

Iterated Ito Integralformula is : $$\int\int...\int dW_{u_1}dW_{u_2}...dW_{u_n}=\frac{1}{n!}t^{\frac n2}h_n(\frac{W_t}{\sqrt t})\\0\leq u_1\leq u_2...\leq t$$and $h_n$ is hermite polynomial of $n^{th}$ degree. I need to prove $$\int_0^t W_sdW_s=\frac{1}{2}(W_t^2-t)$$with respect to above formula. thanks for your help

Answer 1

$$\int dW_{u_1}dW_{u_2}=\frac{1}{2!}t^{\frac 22}h_n((\frac{W_t}{\sqrt t}) $$and $h_2=x^2-1$ take $u_1=u_2$ $$\quad{\int\int dW_{u_1}dW_{u_2}=\int W_{u_1}dW_2\\\frac{1}{2!}t^{\frac 22}h_n((\frac{W_t}{\sqrt t})=\\\int dW_{u_1}dW_{u_2}=\frac{1}{2!}t^{\frac 22}((\frac{W_t}{\sqrt t})^2-1)=\\\frac{1}{2}t(\frac{W_t^2}{t}-1)=\\\frac{1}2(W_t^2-t)\to \int_0^t W_sdW_s=\frac{1}{2}(W_t^2-t)}$$