Determine whether each of the following vector fields $\vec{F}$ is path independent (conservative) or not. If it is path independent, enter a potential function for it, that is, a function $f(x,y)$ so that $\nabla f = \vec{F}$. If it is path dependent, enter NONE.

(a) If $\vec{F}(x,y) = (x + 5) \vec{i} + (3 y + 2) \vec{j}$, then
$f(x,y)$ =

(b) If $\vec{F}(x,y) = (2 x + 3 y)\vec{i} + (3 x + 2 y)\vec{j}$, then
$f(x,y)$ =

(c) If $\vec{F}(x,y) = \langle -3 y, 3x \rangle$, then
$f(x,y)$ =

(d) If $\vec{F}(x,y) = \langle 4 x^2 - y^2, -2xy \rangle$, then
$f(x,y)$ =

(e) If $\vec{F}(x,y) = \displaystyle \frac{3 y}{x+6} \vec{i} + 3 \ln(x+6) \vec{j}$, then
$f(x,y)$ =

(f) If $\vec{F}(x,y) = 2 y \sin(xy) \vec{i} + 2 x \sin(xy)\vec{j}$, then
$f(x,y)$ =