Answer
$F$ is a conservative vector field with potential function $f(x,y)=xe^{xy}+e^y+k$
Work Step by Step
Given: $F(x,y)=(1+xy)e^{xy}i+(e^y+x^2e^{xy})j$
$F=Pi+Qj$ will be conservative when $P_y=Q_x$
$P_y=(2+xy)xe^{xy}$ and $Q_x=(2+xy)xe^{xy}$
Thus, the given vector field $F$ is conservative with potential function $f(x,y)=xe^{xy}+e^y+k$
