9  Surprise

The integral has a closed form in terms of the Meijer G function. Since routines to compute that are included in Mathematica, it is easy to use a standard optimization method to find the answer. We can get 50 digits, but beyond that ran into a problem with the G computation. Bornemann obtained 100 digits. They are:
0.7859336743503714545652439863275455829623954590618668175812318070989103971494123651167706337659944953.

For the solution, we first discover the closed form, and then optimize it.

obj = Simplify[(2 + Sin[10 α]) ∫_1/2^∞ ((2 u - 1)^α Sin[α u])/u^(2 + α) u, α>0]

π^(1/2) Gamma[1 + α] MeijerG[{{}, {(2 + α)/2, (3 + α)/2}}, {{1/2, 1/2, 1}, {0}}, α^2/16] (2 + Sin[10 α])

α /. FindMinimum[-obj, {α, {78/100, 79/100}}, PrecisionGoal12] 〚2〛

0.7859336743

Created by Mathematica  (June 27, 2004)