Answer
$$\phi (x,y) = e^{xy}
$$
Work Step by Step
Given $$\mathbf{F}=\left\langle y e^{x y}, x e^{x y}\right\rangle$$
We need to find $\phi (x,y )$ such that $$ \frac{\partial \phi }{\partial x}= y e^{x y},\ \ \ \frac{\partial \phi }{\partial y}=x e^{x y}$$
By integration we get
\begin{align*}
\phi (x,y)&=e^{xy} +C_1\\
\phi (x,y)&= e^{xy} +C_2
\end{align*}
Then, we can choose $$\phi (x,y) = e^{xy}
$$
