I have the following question, because general physics books like Sears and Serway and others, if not the vast majority, simplify differentials when integrated.
Formally I interpret that this is justified by the first fundamental theorem of calculus, but I'm not sure.
I would appreciate your support, to clarify this doubt, I attach an example.

Let $v=\frac{dr}{dt}$ denote the velocity vector, as let $r$ denote the position vector. Let $F=m\frac{dv}{dt}$ the projection of the net force with the displacement and denote the force at a specified position. Let the mass $m$ be constant. Then the work done along a curve $C$ is defined as,

$$W=\int\_{C} F \cdot dr=m \int\_{C} \frac{dv}{dt} \cdot dr$$

---

The next steps seem sloppy to me.

$$=m \int\_{v\_i}^{v\_f} \frac{dr}{dt} \cdot \frac {dv} {dr} dr$$
HERE, justification for the "cancellation" of the differential
$$=m \int\_{v\_i}^{v\_f} \frac{dr}{dt} \cdot dv$$
$$=m \int\_{v\_i}^{v\_f} v \cdot dv$$

$$=m \int\_{|v\_i|}^{|v\_f|} |v| \,d|v|$$

$$=\frac{1}{2}m|v\_f|^2-\frac{1}{2}m|v\_i|^2$$

$$=\Delta \text{KE}$$