Consider the following integral:
I tried to evaluate I in a closed form (both manually and using Mathematica), but without success.
However, if WolframAlpha is provided with a numerical approximation I≈3.2694067500684…, it returns a possible closed form:
Further numeric caclulations show that this value is correct up to at least 103 decimal digits. So, I conjecture that this is the exact value of I.
Question: Is this conjecture correct?
Sub u=log(2x−1). Then x=log(1+eu)/log2, dx=(1/log2)(du/(1+e−u). The integral then becomes
The nasty pieces of the integral cancel, and we are left with
as correctly conjectured.