P[J_]  :=  Exp[- J J / 2]/Sqrt[2 Pi]

P1[J_]  :=  Integrate[P[K], {K, J, +Infinity}]

Integrand[J_]  :=  8 J P[J] P1[J] P1[J]

DeltaE  =  Integrate[Integrand[J], {J, 0, +Infinity}]

DeltaE // N