The Golden Ratio and the Natural Log: An “Integral” Connection
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- Опубликовано: 1 окт 2024
- Hey there, polymaths!
In today's video video, let's take a look at a definite integral that leads to an absolutely delightful evaluation: the natural log of the golden ratio, phi. It's a journey that starts with the integral of 1/sqrt(x^2+1) from 0 to 1/2.
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I begin by substituting x = tan u and dx = sec^2 u du. This allows us to simplify the integrand to sec u du using a pythagorean identity, and to shift the bounds of integration to 0 and the inverse tangent of 1/2. From there, we're simply integrating sec u du, which you can do by multiplying it by sec u + tan u over itself.
With another substitution, this allows us to integrate and evaluate ln | sec u + tan u | from the previous bounds. Our lower bound evaluates to ln 1, which is 0, and therefore which goes away. Our upper bound turns into ln (sec (tan^-1 (1/2) + tan (tan^-1 (1/2))). The tan and inverse tan cancel out leaving 1/2, and we can set up a right triangle to figure out that sec (tan^-1 (1/2)) is the same as √5/2.
The grand finale? This gives us back ln (1/2 + √5/2), which is of course ln φ, the natural log of the golden ratio.
I did not discover this integral myself, instead I saw David Meyer share his work along with some extensions at: davidmeyer.git.... It turns out you can derive essentially any integral leading to the natural log of an algebraic number by manipulating the bounds and the fact that our original integral is another expression for hyperbolic sine, which has an alternate expression as ln(x + √(x^2 + 1)).
#goldenratio #calculus #integration
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Hi, it’s your student Jaden from a few years ago, just appreciating how much your channel has grown in just a few years! Keep up the great work! Let’s get 100K :)
Jaden, good to hear from you! Yes, the channel's a little more successful than back in you day :). Hope things are well for you and your family!
@@polymathematic Thank you, you too!
I’m in a BC calc class and we have finished everything up to u sub integration by parts partials and diff eqs and slope fields, and although I had to pause it you teach it an such an easy way that you can’t get lost. New sub earned thanks man!
Glad you enjoyed it! Thanks for the sub :)
To me it looks like a kids drawing but to my dad, I remember seeing lots of similar equations on his chalk board at home.
Sadly not even my ADHD could pick up any of it 😮
Nice compact math video 😄
I dislike the use of "tangent inverse" though, since tangent doesn't have an inverse. But the original note used arctan at least!
I created another cool integral like this
(0 to π/2) ∫ ln( 1 + 4sin²(x) ) dx
The answer is πln(φ)
nice! i'll have to play around with that to see if i can get there on my own :)
Have you heard about terrence howard and his 1x1 = 2 "proof"?
i have! i've considered doing a video on it a few times, but i don't know that i can come up with a unique angle.
@@polymathematic First, thanks for taking the time to reply! I'm not sure it's really worth your time. It seems like he's just using a conversational definition for math terms. I just heard about it and I really enjoy your videos and how you explain things.
How serendipitous!