$$ i\hbar \frac{dU_I(t, t_i)}{dt} = \hat{V}_I(t)\hat{U}_I(t,t_i) \tag{10.32} $$ The solutions of this equation, with the initial condition $\hat{U}_I(t_i,t_i)$, are given by the integral equation $$ \hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\int_{t_i}^t\hat{V}_I(t')\hat{U}_I(t',t_i)dt' \tag{10.33} $$
In the derivation of Dyson Series please explain why in equation (10.33) $t$ is changed to $t'$ without integrating.
This is not particular to this problem in QFT but just a mathematical manipulation. The integration is of the form $$\frac{d}{dt} U(t) = f(t)\,\,\,\,\text{with solution}\,\,\,U(t) = U(t_i) + \int_{t_i}^t f(t') dt'$$
Remember the integration variable is dummy - you can have $t$ as the variable and e.g $t’$ as the upper limit but then the $\text{l.h.s}$ will be $U(t’,t_i)$. You can then replace $t’$ with $t$ but this is all done automatically by relabelling the dummy integration variable from the start.
