I came across the YouTube video "I Computed An Integral That Breaks Math" by BriTheMathGuy, where the problem is computing
$$\label{eq_1}\tag{1}\int (x^{dx}-1)$$
And basically to solve this integral we use a little trick writing the argument of $\eqref{eq_1}$ like $$\label{eq_2}\tag{2}\color{red}{\frac{x^{dx}-1}{dx}}\cdot dx$$ The red part of $\eqref{eq_2}$ could be interpreted as $$\label{eq_3}\tag{3}\lim_{h\to 0} \frac{x^h-1}{h} = \log (x)$$ And then we can compute the integral of $$\eqref{eq_1} = \int \log(x)dx = x\log(x) - x + c$$
Is this possible, does $\eqref{eq_1}$ even makes sense to write an integral with $dx$ in that unusual position, or is a glamour social media trick? What disturbs me about $\eqref{eq_2}$ is that even tough we can think of $dx$ as an infinitesimal quantity, deeply it isn't, it is a differential. Has someone some thoughts on this or is just a waste of time? Thank you so much.
You could interpret $\int_{a}^{b}(x^{dx} - 1)$ naturally as a limit of sums $\sum_{i = 1}^{N - 1}(\xi_i^{x_{i + 1} - x_i} - 1)$ where $a = x_1 < \dots < x_N = b$ is a partition of $[a, b]$, $\xi_i \in [x_i, x_{i + 1}]$, and the limit is taken as $\max_{1 \leq i \leq N - 1}(x_{i + 1} - x_i) \to 0$. You could probably recover the result you found rigorously. A similar notation arises in survival analysis for the relation of the survival function to the hazard function.