Why we can operate dx just like a real number in integration?

Question

In physics, we can probably write things like this: ds=vdt. And then we can substitute the expression of ds into the integrand.In mathematics,this is known as integration by substitution. However,during my math lesson, my teacher told me that dx was only a symbol.Then, why can we operate ds like a real number(if dx is not a symbol, it should be an infinitesimal quantity.This differs from real number)? What on earth is the real meaning of dx in integration? Why integration by substitution is just like changing dx to another expression(by exactly following the calculation of real number)? Especially in physics, we actually do not need to consider the phrase "integration by substitution" , but just simply think about the physical meaning of ds=vdt,dm=pdV,dW=Fds ect. and do the substitution. Why we can do these incredible things ?I am very perplexed. SOS! AND THX A LOT FOR YOUR HELP!

Answer 1

Your teacher is right, this is symbolic computation.

It turns out that if you apply to $dx$ some computation rules that look like those on the reals, results are consistent. In fact, mainly the relation to the derivatives

$$dy=\frac{dy}{dx}dx=y'\,dx$$ and the chain rule.

So the notation

$$\int y(x)\,dx$$ is essentially a convenient reminder to handle changes of variable such as

$$\int y(\phi(t))\,dx=\int y(\phi(t))\,\phi'(t)\,dt.$$

But the similarity stops here. Expressions such as $dx+3$ are meaningless.