I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble understanding the integration process in explicitly solving the Hull-White SDE.
So, the Hull-White interest model follows the SDE $$ dR(u) = (a(u) - b(u) R(u)) du + \sigma(u) d\tilde{W}(u) $$ It says the explicit solution can be obtained by applying Ito's Lemma to $$ e^{\int_0^u b(v) dv} R(u) $$ and integrating both sides.
This is where I am having trouble understanding. $$ \int_t^T d\left(e^{\int_0^u b(v) dv} R(u)\right) = e^{\int_0^T b(v) dv} R(T) - e^{\int_0^t b(v) dv} R(t) $$ It seems that we are naively replacing $u$ with $T$ in the first term and with $t$ in the second term. Could we simply do this due to the fundamental theorem of Calculus? Or is there some other working mechanism behind the scenes?