<< Calculus`VectorAnalysis`
SetCoordinates[Cylindrical[r, θ, z]];
  
ϕx[{r_, θ_, z_}]:=k Px r θ (-Sin[θ]);
  
Simplify[Biharmonic[ϕx[{r, θ, z}]]]
  
τrθ = 
      Simplify[D[ϕx[{r, θ, z}], θ]/r^2 - 
          D[ϕx[{r, θ, z}], r, θ]/
            r]
σθ = 
      Simplify[D[ϕx[{r, θ, z}], {r, 2}]]
σrx = 
        Simplify[
          D[ϕx[{r, θ, z}], r]/r + 
            D[ϕx[{r, θ, z}], {θ, 2}]/r^2]
  
ka=k/.Simplify[Solve[Integrate[σrx r Cos[θ],{θ, -α, α}]==-Px,k][[1]] ]
ka = 
      ka /. α -> Pi/2

<< Calculus`VectorAnalysis`
SetCoordinates[Cylindrical[r, θ, z]];
  
ϕy[{r_, θ_, z_}]:=k Py r θ Cos[θ];
  
Simplify[Biharmonic[ϕy[{r, θ, z}]]]
  
τrθ = 
      Simplify[D[ϕy[{r, θ, z}], θ]/r^2 - 
          D[ϕy[{r, θ, z}], r, θ]/
            r]
σθ = 
      Simplify[D[ϕy[{r, θ, z}], {r, 2}]]
σry = 
        Simplify[
          D[ϕy[{r, θ, z}], r]/r + 
            D[ϕy[{r, θ, z}], {θ, 2}]/r^2]
  
ka=k/.Simplify[Solve[Integrate[σry r Sin[θ],{θ, -α, α}]==-Py,k][[1]] ]
ka = 
      ka /. α -> Pi/2

sr[r_, θ_] = (σrx + σry) /. 
        k -> 
          ka
srx = -2 x/
            Pi/(x^2 + 
              y^2)
sry = -2 y/
            Pi/(x^2 + y^2)
ContourPlot[srx,{x, 0.001, 2},{y, -1, 1}]
ContourPlot[sry,{x, 0.001, 2},{y, -1, 1}]

sx = -2 ( 
            x Px +  y Py) x^2/
              Pi/(x^2 + y^2)^2;
sy = -2 ( 
            x Px +  y Py) y^2/
              Pi/(x^2 + y^2)^2;
sxy = -2 ( x Px +  y Py) 
          x y/Pi/(x^2 + y^2)^2;
  
valuex = {Px -> -1, Py -> 0, 
          x -> 1};
Plot[sx /. valuex, {y, -3, 3}]
Plot[sy /. valuex, {y, -3, 3}]
Plot[sxy /. valuex, {y, -3, 3}]

Integrate[x Sin[x]/Cos[x]^3, 
      x]
Integrate[x/Cos[x]^2, x]