Answer
Conservative
Work Step by Step
Given: $F(x,y)=\sin y i+(1+x \cos y) j$
The vector field $F(x,y)=Pi+Qj$ a conservative field throughout the domain $D$, when
$\dfrac{\partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$
Here, $P$ and $Q$ are the first-order partial derivatives on the domain $D$.
Now, $\dfrac{\partial P}{\partial y} =\cos y$ and $ \dfrac{\partial Q}{\partial x}=\cos y$
Hence, the vector field $F$ is conservative.
