Story time
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As an undergrad, I remember seeing the proof for the antiderivative of sec(x).
The key of the proof is to cleverly multiply by sec(x)+tan(x)/sec(x)+tan(x), and pop goes the weasel.pic.twitter.com/Y0iIzvBFFR
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Hmm, I should give that a try myself. I've been doing integrals in more different ways in this class than usual, and learning more about what works and what doesn't.
sec(x) + tan(x) returned itself times sec(x)
There's actually another way to do it that I think would be more instructive than sec(x) + tan(x).. Instead multiply by cos(x)/cos(x), then denominator as 1-sin^2(x).. u-sub sin(x) and pop goes the weasel
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The first time I saw that integral I learned it the sec(x)+tan(x) way. About a year later when I saw the cos(x) way I thought "wait... that's much more intuitive"! Honestly, I think teachers prefer the sec(x)+tan(x) way just to show off.
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It was the only way I knew, except for the approach using complex exponentials (which is ultimately the best, but my calculus students aren't ready for it).
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Hey, I like that... that uses principles I've been teaching them. Usually I do int cos^2(x) dx using a double-angle formula but this time I finally realized you can use integration by parts. Etc.
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which is great, because after a while you begin to see it's all just pulling back 1-forms :)
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