Curvas planas definidas a partir de su curvatura (usando NDSolve)

k[s_] := s Sin[s]

solucion = 
      NDSolve[{theta'[ss] == k[ss], x'[ss] == 
            Cos[theta[ss]], y'[ss] == Sin[theta[ss]], 
          theta[0] == 0, 
x[0] == 0, 
          y[0] == 0}, {theta, x, y}, {ss, -10, 10}]

ParametricPlot[Evaluate[{x[s], y[s]} /. solucion], {s, -10, 10}, 
      AspectRatio -> Automatic]