I was watching a Professor Leonard on U-substitution and he's dividing by $x$ all the time but doesn't that assume that $x \neq 0$ (even though there's no reason why it can't be $0$}
Although it makes the simplification kinda fun because you can see things cancel, is this actually correct? He seems to note at 11:30 that it's different than how others teach and I'm wondering if that's because his way might fail at times?
This is the first example he gives, transforming $\int \left(x^2+1\right)^{50} \cdot 2x dx $ into $\int u^{50} \cdot 2x \frac{du}{2x}$ through this subtitution:
\begin{align*} u = & \ x^2-1 \\ \\ du= & \ 2x dx \\ \\ \frac{du}{dx}= & \ \frac{2x dx}{dx} \\ \\ \frac{du}{2x} = & \ dx \\ \\ \end{align*}
In his substitution, he just cancels the $2x$'s but it seems like he's assuming $x\neq 0$?