Answer
Conservative and $f(x,y)=x \ln y+y \ln x+k$;
Work Step by Step
The vector field $F(x,y)=ai+bj$ is known as conservative field throughout the domain $D$, when we have
$\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}$
$a$ and $b$ represents the first-order partial derivatives on the domain $D$.
From the given problem, we get $\dfrac{\partial a}{\partial y}=\dfrac{\partial b}{\partial x}=
\dfrac{\partial Q}{\partial x}=\dfrac{1}{y}+\dfrac{1}{x}$
Thus, the vector field $F$ is conservative.
Here, we have $f(x,y)=x \ln y+y \ln x+g(y)$
$\implies f_y(x,y)=xy^{-1}+g'(y)$
Here, $g(y)=k$; where $k$ is a constant.
Thus, $f(x,y)=x \ln y+y \ln x+k$
