Answer
$\dfrac{4}{e}$
Work Step by Step
When $F(x,y)=Ai+Bj$ is a conservative field, then throughout the domain $D$, we get
$\dfrac{\partial A}{\partial y}=\dfrac{\partial B}{\partial x}$
$a$ and $b$ are the first-order partial derivatives on the domain $D$.
Here, we have $\dfrac{\partial A}{\partial y} = \dfrac{\partial B}{\partial x}=-2xe^{-y}$
Thus, the vector field $F$ is conservative.
Now, $f(x,y)=x^2e^{-y}+g(y)$ [g(y) : A function of y]
$f_y(x,y)=-x^2e^{-y}+g'(y)$
Here, $g(y)=k$
Thus, $f(x,y)=x^2e^{-y}+k$
Now, $\int_C F \cdot dr =f(2,1)-f(1,0)=(14e^{-1}+k)-(1+k)=\dfrac{4}{e}$