/sci/ - Science & Math

Question for you nerds

In my physics class I ran across a couple instances of integral manipulation that I am not sure how they did legally mathematically.

I get that dx/dt is velocity and that the dx's "cancel out" leaving v*dv but how are they able to just change the bounds of the definite integral to v? The best thing I could come up with is using u substitution but I couldn't get it to work out.

I asked my prof what formal mathematical rule was used and he just said " I dunno, we just do this in physics a lot, not sure what the rule is"