I was wondering about important/famous mathematical constants, like e, π, γ, and obviously the golden ratio ϕ.

The first three ones are really well known, and there are lots of integrals and series whose results are simply those constants. For example:π=2e+∞∫0cos(x)x2+1 dx

e=+∞∑k=01k!

γ=−+∞∫−∞x ex−ex dx

Is there an interesting integral

^{*}(or some series) whose result is simply ϕ?*

Interesting integralmeans that things like+∞∫0e−xϕ dx

are not a good answer to my question.

**Answer**

Potentially interesting:

logφ=∫1/20dx√x2+1

Perhaps also worthy of consideration:

arctan1φ=∫2011+x2dx∫20dx=∫2−211+x2dx∫2−2dx

A development of the first integral:

logφ=12n−1∫F2n+F2n−220dx√x2+1

logφ=12n∫F2n+1+F2n−121dx√x2−1

which stem from the relationship (x−φm)(x−ˉφm)=x2−(Fm−1+Fm+1)x+(−1)m, where ˉφ=−1φ=1−φ and Fk is the kth Fibonacci number. I particularly enjoy:

logφ=13∫20dx√x2+1

logφ=16∫91dx√x2−1

logφ=19∫380dx√x2+1

logφ=112∫1611dx√x2−1

**Attribution***Source : Link , Question Author : Elliptic Curve , Answer Author : Community*