The Golden Ratio and the Natural Log: An “Integral” Connection
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: https://davidmeyer.github.io/qc/golde.... 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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