When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality? Is integration considered a basic operation in the sense you have to do it to all parts of the equation?
$y dy - x dx = 0$
Is it valid to do
$\int y dy - \int x dx = \int 0$
but invalid to leave out a term, for example
$\int y dy - x dx = \int 0$
So do you need to integrate all terms of both sides of the equation?
I find it a little strange because the integration operation may have different integrands so it acts differently e.g. integrating with respect to $y$ instead of $x$. 
In other words does integration work the same way as as any other algebra, like squaring both sides of the equation? 
 A: First, a quick but very important warning:
In basic calculus, the symbol combination
$$ydy - x dx = 0$$
does not really make sense. It is a dirty trick because it is the quickest way of transforming
$$y\frac{dy}{dx} - x = 0,$$
which is a differential equation, into $$\int ydy - \int xdx = 0,$$
which is the next step in solving the differential equation.

That said, yes, when you pull off this dirty little trick, you must always add integrals to all elements which have a $dx$ or $dy$ sign.

Further explanation:
I am trying to say that usually, what happens is this:


*

*You get a task: solve the differential equation $\frac{dy}{dx} y - x = 0$

*You solve the task in two steps. In the first step, you write down $ydy - xdx =0$

*In the second step, you "integrate" what you wrote down and get $\int ydy - \int xdx  = 0$.

*You solve the integrals and extract $y=y(x)$.


In this sequence of steps, writing down $ydy - xdx = 0$ is a step which is technically not correct because the expression $dy$ by itself is not defined, it only makes sense if it is written either in an integral (so $\int ydy$ is well defined) or in a derivative (so $\frac{dy}{dx}$ is well defined). However, the intermediary step (step 2) is a useful trick for rewriting equation (1) into equation (3).
