Integrate[
 PDF[LogMultinormalDistribution[{Subscript[[Mu], 1], Subscript[[Mu],
      2]}, {{Subscript[[Sigma], 1]^2,
      Subscript[[Sigma], 1] Subscript[[Sigma],
       2] [Rho]}, {Subscript[[Sigma], 1] Subscript[[Sigma],
       2] [Rho], Subscript[[Sigma], 2]^2}}], {x, y}]*x, {x, 0,
  Infinity}, {y, 0, Infinity}]

Integrate[(2*Pi*[Sigma]x*[Sigma]y Sqrt[1 - [Rho]^2])^-1*
  Exp[-1/2*(1/(
     1 - [Rho]^2)) (((Log[x] - [Mu]x)/[Sigma]x)^2 -
      2*[Rho]*((Log[x] - [Mu]x)/[Sigma]x)*((
        Log[y] - [Mu]y)/[Sigma]y) + ((
       Log[y] - [Mu]y)/[Sigma]y)^2)]*x, {x, 0, Infinity}, {y, 0, Infinity}]