In order to make the question simpler, I won't give a specific example, but rather a general one:
Suppose I have a DEFINITE integral with boundaries $a$ and $b$ and I use partial integration to get this casual form:
$$\int_a^b udv = u v - \int vdu.$$
As I have boundaries $a$ and $b$, in the end, I have to compute them for the left side( $u*v$ )as well as for the right side ($\int vdu$).
But what if I use substitution when computing $\int vdu$? That should make the boundaries change, according to what is being substituted. Do the boundaries change then for the left side as well or do no changes happen at all? After comparing my solution to a specific task with the solution of an online integral calculator, it seems that no boundaries should change at all. Is that really how it works and why?