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Yearly, several my 17 year old calculus students always asks, how can they foreknow this trick to multiply with $\color{red}{\dfrac{\cos x}{\cos x}}$? How can I demystify, that "the way I used to integrate $\dfrac 1{\color{limegreen}{\cos x}}$ is to multiply with $\color{red}{\dfrac{\cos x}{\cos x}}$" ?

$\int \dfrac1{\cos x} dx = \int \dfrac{\cos x}{\cos^2 x} dx =\int \dfrac{d(\sin x)}{1 - \sin^2 x}$

"If one were developing the theory from scratch, how would one find this solution (other than blind luck)?" I seek "good explanations and motivations for everything (as opposed to just pulling out ready-made solutions like what was done to me when I was learning this exact thing)".

NOT asking about other integrations like integrating any function of the form $\sin^n(x)\cos^m(x)$,
Tangent Half-Angle Substitution, or
multiplying by $\color{limegreen}{\sec x} \dfrac{\color{goldenrod}{\sec + \tan}}{\color{goldenrod}{\sec+ \tan}}$.

user13772
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    Please don't use such long links. They are supposed to stand out from the rest of the text. – preferred_anon Nov 01 '23 at 23:41
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    I am not sure I would expect someone in their first calculus class to intuitively find any of the tricks to integrate $\sec x$ or $\sec^3 x.$ Rather, if I were teaching it, I would say someone clever found it, and show that it indeed works. – user317176 Nov 01 '23 at 23:47
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    FYI, this blog post gives a history (and links to others) of this "integral that took 100 years to solve", noting that the result was approximately observed in 1569 by Mercator while devising his maps, then exactly guessed in 1645 by Henry Bond who happened to notice a pattern in some logarithm tables (back when everyone had those), all well before James Gregory proved it via the then-still-new-fangled Calculus in 1668. And yet, it still took a couple of years before Isaac Barrow devised a clever approach. – Blue Nov 02 '23 at 00:01
  • I don't think first-time calculus learners can naturally come up with the $\frac{f(x)}{f(x)}$ trick easily. It just comes down to trying out some manipulations and seeing if they work, sort of like how in integration by parts, you guess what to differentiate and what to integrate. – Accelerator Nov 02 '23 at 00:03
  • Possibly useful: Ways to evaluate $\int \sec \theta , \mathrm d \theta$ and these other questions (maybe this and most of the others are already cited by you -- I didn't look at all your links). – Dave L. Renfro Nov 02 '23 at 00:11
  • @DaveL.Renfro I linked to, and perused, them already. I wrote that I NOT seeking other methods or solutions. – user13772 Nov 02 '23 at 00:12
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    Maybe also What's the most effective way to introduce/motivate the anti-derivative of $\sec x$?. I linked to, and perused, them already. --- FYI, it helps to give the exact title of the questions you're linking to in a situation like this, so others can easily tell which previous questions you have cited. – Dave L. Renfro Nov 02 '23 at 00:17
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    Is there even a question here? I'm sorry, this post is a bit hard to follow. – Sean Roberson Nov 02 '23 at 00:18
  • @DaveL.Renfro I linked to that post too. Sorry, English not my first language. You have go-ahead to edit my question. – user13772 Nov 02 '23 at 00:20
  • @SeanRoberson Sorry, English not my first language. You have go-ahead to edit my question. – user13772 Nov 02 '23 at 00:20
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    I think the question is pretty clear. "Is there a heuristic that would make this particular trick easier to think of before-hand?" – Jair Taylor Nov 02 '23 at 00:37
  • @SeanRoberson Jair Taylor understood my question perfectly. – user13772 Nov 15 '23 at 00:14
  • @JairTaylor Thanks for the support! You understood my question perfectly. You have my go ahead to edit and improve my question, if you like. – user13772 Nov 15 '23 at 00:14

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I think it applies to a lot of areas of mathematics, but not everything necessarily has an intuitive approach from first principles that someone might be expected to notice. Sometimes people stumble upon things by experimentation, or by being very clever and noticing patterns/relations, and so on. Once you "know the trick" it becomes a new tool you can use even if it doesn't have an obvious motivation.

This particular integral even gets mentioned as a specific example of accidental discoveries in Spivak's calculus:

There is another expression for the integral of sec x dx, which is less cumbersome than $\log(\sec x + \tan x)$; using Problem 15-9, we obtain the integral as $\log(\tan(x/2 + \pi/4))$. This last expression was actually the one first discovered, and was due, not to any mathematician’s cleverness, but to a curious historical accident: In 1599 Wright computed nautical tables that amounted to definite integrals of $\sec$. When the first tables for the logarithms of tangents were produced, the correspondence between the two tables was immediately noticed (but remained unexplained until the invention of calculus).

So in this case, even the first discovered answer is still useful information as it now gives you an endpoint for other transformations and substitution experiments in terms of direction. But sometimes discoveries are just accidents and any "intuitive" explanation is going to come after the fact with that pre-knowledge driving it.

user1947180
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    If I remember correctly, the $\log\tan$ version is what falls out if you do a Weierstrass substitution. I didn't post that as an answer, because I still consider it a trick, but it doesn't require any insight in the derivation - you make the substitution and follow your nose. – preferred_anon Nov 02 '23 at 09:18