Typically in stochastic integrals, it is common to have integrals with respect to the Wiener increment $dW(t)$.
For example
$$ \int_{0}^{t}W(t)dW(t) = \frac{W^{2}(t)}{2}-\frac{t}{2} $$
by Ito's lemma. But, how does one go about computing
$$ \int_{0}^{t}W(t^{\prime})dt^{\prime} $$
? I know that $dW(t)^{2}\equiv dt$ but I am not sure how to proceed integrating
$$ \int_{0}^{t} W(t^{\prime})\sqrt{dW(t^{\prime})}. $$
What should be the approach here?
As in the answer of your another question, you have $$\begin{align} \int_0^tW_t'dt' &= \int_0^t(t-u)dW_u \\ \end{align}$$ and then you can determine easily the law of this integral $$\int_0^tW_t'dt' \sim \mathcal{N}\left(0,\int_0^t(t-u)^2du \right)$$