Why is $int x^2e^{x^2},d(x^2)$ not proper notation?

Question

So, as the title states. My question is about mathematical notation. Whenever my lecturer in calculus one computes an integral they do some in my opinion very redundant steps.
When they attempt to compute for example $int x^3e^{x^2},dx$, they will write down $u = x^2; du=2xdx$ and proceed to turn the integral into $frac{1}{2}int ue^u,du$.
I don't see why it is required to hide this simple mathematical truth that $d(x^2)=2xdx$ behind a substitution, yet they tells me that not doing so would technically be incorrect.
By the way, I am not asking this to proof my lecturer wrong, since they are an epic person and when I told them that I found it easier without the substitution I was given permision to do so. I am merely asking why it would be "technically incorrect" to do so.

Mark Viola · Accepted Answer

The notation is proper.  The Riemann-Sieltjes for a real-valued function $f(x)$ for $xin [a,b]$ with respect to a real-valued function $g(x)$ is written
$$I=int_a^b f(x) dg(x)$$
If $g$ is differentiable on $[a,b]$, then $I$ can be written as
$$I=int_a^b f(x)frac{dg(x)}{dx},dx$$
However, $g(x)$ need not be differentiable and, in fact, can even be discontinuous.
In the case of interest, $g(x)=x^2$ and $f(x)=x^2e^{x^2}$.  We have, therefore
$$int_a^b x^2e^{x^2}d(x^2)=2int_a^b x^3 e^{x^2},dx$$

Yves Daoust · Answer

Students can feel more comfortable when they see what is substituted as a single entity. But of course
$$int x^3e^{x^2}dx=frac12int x^2e^{x^2}d(x^2)=frac12int ue^{u}du$$
are equivalent and technically correct.
You can even do the integral mentally if you want (but make sure your instructor believes you).

Extra trick:
Write down $x^2$ on a label and read it loud "$u$".

Why is $int x^2e^{x^2},d(x^2)$ not proper notation?

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