A fabulous integral from MIT Integration Bee | infinite nested roots

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[@mikecaetano]

How interesting that the natural log of the golden ratio popped out as an upper boundary during a substitution considering that the transformed integrand takes on the continuous form of the metallic ratio.

[@ahmedgg8867]

nice integral you are the best keep going

[@thiagocabre631]

Great video!
Quick question.
Can't you substitute u for (x+1) and then express the infinite nested roots as a sum of powers, creating a geometric sequence with u1 = 1/2 and r = 1/2. This can be expressed as the infinite sum of a geometric sequence which is equal to u^1. Then you integrate udu which gives (u^2/2). Substitute u for (x+1), apply the boundaries giving the result (3/2)?
Thanks again.

[@whenlifeisintoxicated3048]

I don't know, why people complicate matters.
The problem itself indicates,
Substitute , x=y-1/y
dx=(1+1/y^2)dy
I=int[from 1 to gr ](y+1/y)dy=(gr^2-1)/2+ln(gr)

[@groovygames4313]

yeah i, too, feel this was overly complicated. this substitutions works well. or just solve that quadratic formula type integrand with that (x^2+a^2)^1/2 formula. got the answer in 40 seconds💀

[@bjhhar1233]

woo