I ran across an interesting integral and I am wondering how in the world it could relate to the Golden Ratio, $\frac{1}{\phi}$.
The problem says the solution must include the Golden Ratio, $\frac{1}{\phi}=\frac{\sqrt{5}-1}{2}$.
$\int\limits_{-\infty}^{0}n^{x}(n+1)^{x}dx$
I evaluated it easy enough using parts. I arrived at
$\frac{1}{ln(n^{2}+n)}$.
But, it escapes me is how this can be written in terms of the aforementioned Golden Ratio.
I found something in Excursions in Calculus by Robert Young that relates a logarithmic spiral to the Golden Ratio, but it seems rather iffy.
$\frac{1}{ln(\beta)}=\frac{\pi}{2ln(\phi)}$
Replacing beta with $n^{2}+n$ and solving for n gives a solution, but I doubt if it is correct.
I notice that n and n+1 could be somehow related to the Fibonacci sequence?.
Any thoughts are appreciated. Thanks