Answer
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
Work Step by Step
$\textbf F(x,y) = e^x (\sin y \textbf i + \cos y \textbf j )$
$\nabla \times \textbf F =\begin{vmatrix}
\textbf i & \textbf j & \textbf k\\
\frac{\delta}{\delta x} & \frac{\delta}{\delta y}& \frac{\delta}{\delta z} \\
e^x\sin y & e^x\cos y & 0\\
\end{vmatrix} = 0\textbf i + 0\textbf j + (e^x\cos y - e^x \cos y)\textbf k = 0
\\$
As $\nabla \times \textbf F = 0$, so, $\textbf F$ is conservative
