My question is very basic, and maybe I'm just having a brain malfunction, but I'm curious why it's ok to integrate both sides of an equation. It's pretty easy to integrate a lot of functions, but there are a lot of operations implicitly being performed in the background of any integration. So I'm just wondering how we know that, when integrating both sides of an equation, the equality of the sides is preserved. In a related vein, I'm having trouble understanding the types of algebraic manipulation that can be performed on differentials. For example, I've viewed a number of Khan Academy videos where Sal pretty casually multiplies both sides of an equation by dy or dx. This makes sense to me. However, I have read that this is an abuse of notation. Thank you.