Use Green's theorem to evaluate $\int_{C} F \cdot d r$. (Check the orientation of the curve before applying the theorem.) \[ \int_{C}\left(3+e^{x^{2}}\right) d x+\left(\tan ^{-1}(y)+3 x^{2}\right) d y \] Need Help? Read It

Step 1 ：Given the vector field \(F = P(x, y)i + Q(x, y)j\), where \(P(x, y) = 3 + e^{x^2}\) and \(Q(x, y) = \tan^{-1}(y) + 3x^2\).

Step 2 ：We need to find the curl of F, which is \(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\).

Step 3 ：Calculating the curl of F, we find that it is \(6x\).

Step 4 ：We then evaluate the double integral of the curl over the unit square D. The result of this calculation is 3.

Step 5 ：By Green's theorem, this is the value of the line integral of F over the curve C.

Step 6 ：Final Answer: The value of the line integral is \(\boxed{3}\).