Answer
$F$ is a conservative vector field and $$\int_CF.dr=0$$
Work Step by Step
$F(x,y)=(4x^3y^2-2xy^3)i+(2x^4y-3x^2y^2+4y^3)j$
Since, $F=Pi+Qj$ will be conservative when $P_y=Q_x$
Thus, $$P_y=8x^3y-6xy^2$$
and $$Q_x=8x^3y-6xy^2$$
This shows that the given vector field $F$ is conservative.
By the fundamental theorem of line integrals, we have
$\int_CF.dr=f(1,1)-f(0,1)$
$=(1-1+1+k)-(0-0+1+k)$
$=0$
Hence, the result is: $\int_CF.dr=0$
