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