I made my question less abstract in hopes it makes sense to someone.
If you look at the antidervative of g over a given interval (say -1,0), then you have g(t)=-10, which has -10x as an antiderivative
So over this interval,
So no you can't write the last part you did. It would be like you integrate from 0 to x, which is false !
It's easy to see that on a graph, since the integral is the area under the line at y=-10
I don't know if it's clear... ?