Answer
$F$ is a conservative vector field and $\int_CF.dr=2$
Work Step by Step
Given: $F(x,y)=e^yi+(xe^y+e^z)j+ye^zk$
$F=Pi+Qj+Rk$ will be conservative when $R_y=Q_z$,$P_y=Q_x$, and $P_z=R_x$
and $R_y=Q_z=0$,$P_y=Q_x=0$, and $P_z=R_x=0$
Thus, the given vector field $F$ is conservative.
By the fundamental theorem of line integrals
$\int_CF.dr=f(4,0,3)-f(0,2,0)=2$
Hence, $\int_CF.dr=2$
