Integrate[
   phi[x] Log[phi[x]] + (1 - phi[x]) Log[1 - phi[x]] +
     chi phi[x] (1 - phi[x]) + (1/2) phi'[x]^2,
   {x, 12, 20}]

chi = 62/27;
phi[x_] := x Log[x] + (1 - x) Log[1 - x] + chi x (1 - x)