{2 x Cos[(x^2) + (z^2)], Sin[(x^2) + (z^2)], 
      2 z Cos[(x^2) + (z^2)]}

{,,D[Sin[(x^2) + (z^2)], x] - 
        D[2 x Cos[(x^2) + (z^2)], y]}

{2 x Cos[(x^2) + (z^2)], Sin[(x^2) + (z^2)], 
      2 z Cos[(x^2) + (z^2)]}

Curve

{Cos[t], t, Sin[t]}

Integrate[{2 Cos[t] Cos[1], Sin[1], 
          2  Sin[t] Cos[1]} . {-Sin[t], 1, Cos[t]}, {t, 0, 
        2 Pi}]

Start

{Cos[0], 0, Sin[0]}

End

{Cos[2 Pi], 2 Pi, Sin[2 Pi]}

Curve

{1, t, 0}

Integrate[{2 1 Cos[1 + 0], Sin[1 + 0], 2 0 Cos[1 + 0]} . {0, 1, 
          0}, {t, 0, 2 Pi}]

Let us try the curve

{t, t^2, 0}

from (0,0,0) to (1,1,0)

{2 x Cos[(x^2) + (z^2)], Sin[(x^2) + (z^2)], 
      2 z Cos[(x^2) + (z^2)]}

Integrate[{2 t Cos[(t^2) + 0], Sin[(t^2) + 0], 
          2 0 Cos[(t^2) + 0]} . {1, 2 t, 0}, {t, 0, 1}]

Let us try the curve

{t, (t^4), 0}

from (0,0,0) to (1,1,0)

{2 x Cos[(x^2) + (z^2)], Sin[(x^2) + (z^2)], 
      2 z Cos[(x^2) + (z^2)]}

Integrate[{2 t Cos[(t^2) + 0], Sin[(t^2) + 0], 
          2 0 Cos[(t^2) + 0]} . {1, 4 (t^3), 0}, {t, 0, 1}]

Problem 28 in 18.4

ParametricPlot[{((3 t)/(1 + (t^3))), ((3 (t^2))/(1 + (t^3)))}, {t, 
        0, 100}, PlotPoints -> 200, 
      AspectRatio -> Automatic]

{0, x}

Simplify[
      D[{((3 t)/(1 + (t^3))), ((3 (t^2))/(1 + (t^3)))}, t]]

Integrate[{0, ((3 t)/(1 + (t^3)))} . {((3 - 6 (t^3))/(((1 + (t^3)))^2)), -((3 t (-2 + (t^3)))/(((1 + (t^3)))^2))}, {t, 
        0, Infinity}]