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