Integrate with $-d(x/y)$ Here's an integral which I encountered that uses some unfamiliar notation for me:
$$\int-\frac{d(x/y)}{\sqrt{1+(x/y)^2}}$$
What does this mean? I don't  have much of an idea. 
Edit: This problem is from a book on differential equations ($y$ is a function), and the author writes:

$$-\frac{d(x/y)}{\sqrt{1+(x/y)^2}}=\frac{dx}{x}$$ Integration of this now gives $$-\log\left|\frac xy +\sqrt{1+(x/y)^2}\right|=\log|x|+\log|c|$$

How do you get to this step?
 A: Multiple choice:


*

*$y$ is a constant, i.e. something that doesn't change as $x$ changes, and you're antidifferentiating a function of $x$.  Then this becomes $$ -\int\frac{du}{\sqrt{1+u^2}}. $$

*It's not altogether impossible that $x$ is a constant and you're antidifferentiating a function of $y$, so that $\displaystyle d\left(\frac x y \right) = \frac{-x}{y^2}\,dy $

*$x$ and $y$ are related to each other in some way specified elsewhere than in what is posted in the question above.  If, for example, both are functions of $t$, then we could have $$d\left(\frac x y \right) = \frac{y\,dx-x\,dy}{y^2} = \frac{yx'-xy'}{y^2}\,dt,$$ where $x'$ means $dx/dt$ and $y'$ means $dy/dt$.

*Maybe something else?


(The third bullet point above actually includes both of the first two.)
I wouldn't use such notation without at least having some preceding context that would make it clear which choice above is right, and even then maybe I wouldn't be too comfortable with it.
A: The notation is a bit unclear to me. But let's say that you want to find the following integral.
$$
\int -\frac{1}{\sqrt{1 + (x/y)^2}}\; dx.
$$
Here, from the notation, we assume that $y$ is a constant. The standard approach is integration by substitution. So we let $u = x/y$ Then $du = (1/y) dy$. So we get the integral
$$
\int -\frac{1}{\sqrt{1 + (u)^2}}\; y\;du.
$$
This we can find. (something with an inverse trig. function).
Another notation for the above is the notation that you ask about. It is, however, a bit more unclear in my opinion.
Anyway, the point is that this notation is an "alternative" to integration by substitution.
