Okay, so this is a slightly tricksy problem, because at first people think "Oh, it's fine, I just multiply through by this thing and it's fine" but then they think "Hang on, we've never actually defined what we meant by this... it's just some shorthand trickery" - but then if they eventually do differential geometry you think "Aha! So it did make sense all along!"
The main message I want you to take away is that things like $\mathrm d x$ are actually well defined things called differential forms which you don't really need to understand in any detail at all to get how they work.
The way they end up working in integration and changes of variable is roughly the following: they come together to make a volume form which just tells you how much volume a small range of your parameters corresponds to. (I say "come together" because if you are doing many integrations like in $\int \int f(x,y) \;\mathrm d x \mathrm d y$ then you get a bunch of the $\mathrm d [...]$ things all together.) More precisely, remember how you can roughly define integration as a limit of a sum like
$$\int_a^b f(x) \mathrm d x \equiv \lim_{N\to\infty}\sum_{n=1}^N f(x_n) \left(x_n-x_{n-1}\right)$$
where $x_0=a,x_N=b$ and the other points $x_i$ are chosen in between, such that all gaps $\delta_n = x_n-x_{n-1} \to 0$ (say, uniformly) as $N\to \infty$.
Here, $\delta_n = x_n-x_{n-1}$ is providing some measure of how important the bit of space between $x_{n-1},x_n$ is in computing the integral. The $\mathrm d x$ is what keeps track of that information.
Suppose you then try $u=x^2$ or $x=\sqrt u$. Then in general
$$\int_{a^2}^{b^2} f(\sqrt u) \mathrm d u \equiv \lim_{N\to\infty}\sum_N f(\sqrt{u_n}) \left(u_n-u_{n-1}\right) = \lim_{N\to\infty}\sum_N f(x_n) \left(x_n^2-x_{n-1}^2\right)\neq \int f \mathrm d x$$
because the weight is different!
But notice that $x_n^2-x_{n-1}^2 = (x_n-x_{n-1})(x_n+x_{n-1}) \approx (x_n-x_{n-1})(2x_n)$ in the limit of fine spacing, so
$$\int_{a^2}^{b^2} f(\sqrt u) \frac{\mathrm d u}{2x} = \int_a^b f(x) \; \mathrm d x$$
We're really analyzing the difference between the volumes of the little patches of space when we play with the differentials. The trick is to realize that in general, just like here, $\mathrm d u = u'(x) \mathrm dx$. In higher dimensional integrals, you will discover that the generalization to e.g.
$$\int f(x,y) \;\mathrm d x\mathrm d y = \int f(u,v) \; J \; \mathrm d u\mathrm d v$$
where $u=u(x,y),v=v(x,y)$ involves a quantity $J$ called the Jacobian (determinant) which uses all the possible derivatives of $u,v$ with respect to $x,y$ in a particular way.
The notation
$$\frac{\mathrm d u}{\mathrm d x} = \lim_\text{fine spacing}\frac{\delta u}{\delta x} = \lim_{x_n-x_{n-1}\to 0}\frac{u_n-u_{n-1}}{x_n-x_{n-1}} = u'(x)$$
is now seen to be just a suggestive notation which works for the case of only one variable changing. It's used because it makes it clear how the volume form should be replaced.
When there are many variables, this notation breaks down because the factors are all mixed up together and people write partial derivatives, which you'll see soon if you haven't already, instead. It turns out that it makes sense to use a generalization of the
$$\mathrm d u = u'(x) \mathrm d x$$
law called the chain rule in which, for $u=u(x,y)$ for example
$$\mathrm d u = u_x \mathrm d x + u_y \mathrm d y$$
where $u_x(x,y)$ is the derivative of $u$ with respect to $x$ when we just think of $y$ as a constant.
You'll have to wait until differential geometry courses to see how to use this to get the Jacobian factor; it turns out that rather than just writing the forms together, you should technically define something called a wedge product such that $a\wedge b = -b\wedge a$ for one-forms like $\mathrm d x$; then you get
$$\mathrm d u \wedge \mathrm d v = (u_x \mathrm d x + u_y \mathrm d y)\wedge(v_x \mathrm d x + v_y \mathrm d y) = (u_x v_y-u_y v_x) \mathrm d x \wedge \mathrm d y$$
so that the Jacobian is (one over) $(u_x v_y-u_y v_x)=\det \pmatrix{u_x & u_y \\ v_x & v_y}$.
You can get this result directly from thinking about little patches of volume, however, so you'll see this far earlier than any differential form stuff. I just thought that, since you were curious, you should have had the full story mentioned to you along the way.
