Thank you, Professor Rogers. As with your answer to my previous question, you have provided quite useful information to me and I am grateful. I appreciate your style, and I am making notes-to-self about it for future use.

Rergarding textbooks and Substitution, I am primarily using the three volumes of "Calculus":

1.5 | Substitution https://openstax.org/books/calculus-volume-2/pages/1-5-substitution

In selecting this text I based my decision on 1) the heavy contribution of Gilbert Strang -- 29 years at MIT -- and 2) the fact that it is an online pdf which can be broken down by chapter/section and loaded on my new iPad, thus allowing me to work in bed (or LazyBoy), and happily eliminating pencils and paper.

(restating in my words):

The general topic of "Integration By Substitution" can be divided into the forms $u = h(x)$ and $x=k(u)$.

The former case, $u = h(x)$, applies a set of complete set of consistent, simultaneous substitutions to the integrand formula. These substitutions can be variable renames and sub-expression regroupings in the formula parse tree which do not alter the output value of the formula or the independent variable, $x\$. In this type of substitution, the elements comprising $du$ must be present in the original integrand -- they are not added as part of the substitution, though conceivably they could be modified by the substitution.

The latter case, $x=k(u)$, actually redefines the independent variable ( $x$), and after applying the complete, consistent set of substitutions to $x$, the differential $dx$ is recomputed based on the new definition of $x$... and inserted in the integrand.

Re: the example from W|A, I think Step 2's verbiage could be improved by clarifying that the substitution, $u = e^x$ applies only to the radical sub-expression, not the entire integrand. I would be curious to learn how Mathematica's code handles it.