I was following the proof of Noether's theorem in Lemos - Analytical Mechanics, page 73.
He considers a full infinitesimal transformation: $$t'=t+\epsilon X(q(t),t),$$ $$q'(t')=q(t)+\epsilon\Psi(q(t),t),\tag{2.160}$$ whose change in the action is $$\Delta S=\int_{t_1'}^{t_2'}L\left(q'(t'),\frac{dq'(t')}{dt'},t'\right)dt'-\int_{t_1}^{t_2}L\left(q(t),\frac{dq(t)}{dt},t\right)dt.\tag{2.161}$$ Note that integration limits are changed in the first term of the RHS.
Then after plugging the transformation in $\Delta S$ he gets $$\Delta S=\int_{t_1}^{t_2}L(q+\epsilon\Psi,\dot q+\epsilon \xi,t+\epsilon X)(1+\epsilon\dot X)dt-\int_{t_1}^{t_2}L(q,\dot q,t)dt,\tag{2.166}$$ where $$\xi=\dot\Psi-\dot q\dot X.\tag{2.165}$$
Why is the first integral above over $[t_1,t_2]$ instead of $[t_1',t_2']?$
Is not there a term proportional to $\epsilon\left[L(q(t_2),\dot q(t_2),t_2)-L(q(t_1),\dot q(t_1),t_1)\right]$ being neglected in the $\Delta S$ above?