Using Ito's Lemma to derive $\int^{T}_0 W_tdW_t =\frac{1}{2} W_{T}^2-\frac{1}{2} T$ where $W_t$ is brownian motion and $W_t=0$

Question

Using Ito's Lemma, derive $$\int^{T}_0 W_tdW_t =\frac{1}{2} W_{T}^2-\frac{1}{2} T$$ where $W_t$ is brownian motion and $W_t=0$

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Appreciate a hint ; dont know where to start.

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Answer 1

Hint: try applying Ito's lemma to $X_t=f(t,W_t)$ for $f(t,x)=\frac{1}{2}x^2$.

Answer 2

$$f(W_T)=\int_{0}^{T}f'(W_t)dW_t+\frac 12 \int_{0}^{T}f''(W_t) dt$$ Set $f(x)=x^2$ thus $$W_T^2=2\int_0^T W_t dW_t+ \int_0^T dt=2\int_0^T W_t dW_t + T$$