Answer
$0$
Work Step by Step
For a vector field to be Conservative, $\dfrac{\partial f_1}{\partial y}=\dfrac{\partial f_2}{\partial x}$
We are given that the force field as: $F(x,y)=(e^{-x} \cos y, \sin y)$
So, we can see that $\dfrac{\partial f_1}{\partial y}=-e^{-x} \sin y=\dfrac{\partial f_2}{\partial x}$
Next, we will find the potential function for the given vector field as:
$\phi(x,y)=-e^{-x} \cos y$
Therefore, the integral can be expressed as:
$\int_C F \ dr=\phi(1,1) -\phi(1,1) \\=-e^{-1} \cos (1)-(-e^{-1} \cos 1)\\=0$
