I = ∫[0,π] x・sin(x)/[2−cos(x)^2] dx　　(置換 x = π−t)
　= ∫[0,π] (π-t) sin(t)/[2−cos(t)^2] dt

(第1式 + 第2式)/2 より
I　= (π/2)∫[0,π] sin(x)/[2−cos(x)^2] dx　(置換 u=cos(x))
　= (π/2)∫[-1,1] du/(2-uu)
　= (π/(4√2))∫[-1,1] {1/(√2 +u) + 1/(√2 -u)} du
　= (π/(4√2))[ log|(√2 +u)/(√2 -u)| ](u：-1→1)
　= (π/√2) log(1+√2)
　= 1.9579198…