What does it mean when people interact with the $dx$ at the end of an integral like it's a number?

Question

I was always taught that the $dx$ at the end of an integral is just notation that 1) tells you which variable to integrate and 2) acts as an end to the integration like a full stop (period).

But I see people putting the $dx$ on fractions, moving it to the other side of the equation, and other stuff acting like it's a number. You wouldn't interact with the notation of, say, $\sin(x)$ in this way. It seems rediculous the thought of interacting with the notation as if they were numbers like saying the "sin" can be moved to the other side in $\sin(x)=3$ resulting in $x=3/(\sin)$. SO why can you interact with the $dx$? Thanks, by the way!

Answer 1

Formally speaking, you are correct. Typically, when people are moving $dx$ terms around they are being loose with the notation, but it is usually in support of applications of integration by substitution (aka change of variable).

For example, if I write

$$\begin{eqnarray} x & = & \sin u \\ dx & = & \cos u\ du \end{eqnarray}$$

what I'm really doing is preparing for the substitution in something like

$$\begin{eqnarray} \int \sqrt{1 - x^2} dx & = & \int \sqrt{1 - \sin^2 u} \frac{dx}{du} du \\ & = & \int |\cos u| \cos u\ du \end{eqnarray}$$

This is used in more formal contexts as well, for example the canonical form of an exact differential equation is

$$P\ dx + Q\ dy = 0$$

even though technically it should probably look something more like $P + Q \frac{dy}{dx} = 0$

Answer 2

Some additions to the good accepted answer.

Intuitively, think of "$dx$" as "a small change in $x$". That makes sense in many contexts. In the integral $$ \int_a^bf(x)dx $$ the integrand $f(x)dx$ is the area of the very thin rectangle with base $dx$ and height $f(x)$. Leibniz chose the integral sign to suggest a sum: the area is the sum of all the infinitesimally thin rectangles.

The expression $$ dy = f'(x)dx $$ says that to calculate the small change in $y$ caused by the small change $dx$ in $x$ you multiply $dx$ by the slope $f'(x)$. Alternatively, the ratio of those small changes is the derivative.

These manipulations with infinitesimal quantities are not rigorous as I've described them. They can be made rigorous. But even if you don't go as far as learning that rigor, you can develop your intuition and then use them to guess and remember results. Physicists and mathematicians do this correctly all the time.