Show that

$F(x,y,z)=(z^{3}cosx+2x{y}^2)i+(2x^{2}y-2)j+(3z^{2}sinx-4)k$

represents a conservative field. Find a scalar potential function $,$ such that $F=$grad$.$ Hence, evaluate

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F.dr,$

where $C$ is is any path joining point $(0,-1,1)$ to $(2,3,6)$ and then to $(\frac{}{2},2,2).$