This picture is from Neftci's textbook, 'An Introduction to the Mathematics of Financial Derivatives, Third Edition'
What makes me uncomfortable is equation [10.61] In above picture. In this equation,$sW_s$ in the $d[sW_s]$ is definitively stochastic term.
So I think It is not rigorous to apply Riemann integral in this term. I mean, $∫^t_0dX = [X]^t_0 = t - 0... So ∫^t_0d[sW_s] = [sW_s]^t_0 = tW_t - 0*W_0. (because of W_0 = 0)$ this fundamental relationship should not be applied, because it has stochastic term.
But In the above picture, equation [10.61] seems to apply Riemann integral property not Ito integral property.
I want to know the reason why.