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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So I started to play around and realized that this factor we multiplied by actually fulfilled a specific property.
sec(x) + tan(x) returned itself times sec(x)
I gave this property the name "quasi-eigenfunction"
I wondered if this could be applied in more general settings..pic.twitter.com/QN1U2FWXxE
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A year later I got to differential equations. The professor showed that what I called quasi-eigenfunctions is called an integrating factor, and spoke about its use. When I found this out, I felt a bond of sorts to past mathematicians; a kinship over centuries. It was wonderfulpic.twitter.com/pZ8h9iaPZx
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Replying to @InertialObservr
I have to teach my calculus students how to integrate sec(x) pretty soon, maybe this week. It's a bit traumatic since it must make them think "how the hell could anyone have guessed that?"
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Replying to @johncarlosbaez
That's true.. I know there's another way, but it's probably more traumatic. Write cosine as complex exponentials --> u=e^{ix} --> partial fractions.. Yea, they'll love that.
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Replying to @InertialObservr
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.
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Replying to @johncarlosbaez
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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Replying to @InertialObservr @johncarlosbaez
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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Replying to @wood_croft @InertialObservr
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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Replying to @johncarlosbaez @InertialObservr
On the other hand... the sooner they get used to complex numbers the better.
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Replying to @wood_croft @johncarlosbaez
You don’t want to overwhelm them needlessly
7:48 PM - 13 May 2019
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