d[y_] := 
      (1/2) + (1/π) ArcTan[(1/4) (1 + (1/10000) 
                    Exp[y - 78]) (y - 78)]

Plot[d[y], {y, 0, 120}, 
      PlotRange -> {{0, 120}, {0, 1}}]

lifeexp = NIntegrate[y d'[y], {y, 0, 150}]   // N