The main question of this post is, perhaps, of the "homework" variety but maybe my request stems from either a) unclear or inadequate explanations in WolframAlpha Pro and Mathworld (and other sources of knowledge); or b) my failure to comprehend the simplest instructions. And it seems I need to find a tutor, if anyone wants a job.
There are two different kinds of substutution involving differentials in integrals. These are illustrated in Steps 2 and Steps 3 from the WolframAlpha Pro request+snapshot:
Step 2 substitutes u = e^t and du = e^t dt but u is substituted only inside the root, and the e^t on the outside is assumed to be part of the differential, du. In fact, if the e^t on the outside of the root did not exist then the substitution would be impermissible (?)
Step 3 is different, a trigonometric substitution which happily generates the correct differential, du = 2 sec^2(s) ds from the original dx.
So, what distinguishes these substitutions?
Both of these forms of substitution are found in standard calculus textbooks, including free online ones, if you wish to consult them
The most general form of substitution, which is not usually found in textbooks, is for an integral of the form
$\int f(x) \; dx$, choose a relation (equation)
$g(x,u)=0$ and use it and its differential to eliminate
$f(x) \; dx$. The two commons forms for the relation are
$u = h(x)$ and
$x = k(u)$, corresponding to the
$e^x$ and the trigonometric substitutions respectively in the W|A output. The first form
$u = h(x)$ is usually the first type of substitution introduced in calculus books. I think this is because guessing what to choose for
$h(x)$ is fairly straightforward in many integrals. The second form tends to be used in special circumstances, such as trigonometric substitution. If
$k$ is invertible, then one form can be converted to the other.
As an example of substitution using the general
$g(x,u)=0$, note that the integral at hand may be solved by using
$e^x - 2\tan\theta = 0$ to eliminate
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
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.