Vector potential of a 2 dimensional field

1. Homework Statement
Compute vector potential for the following field:

$$\vec{F}$$ = <xy, y^2/2 >

2. Homework Equations

$$^{0}_{1}$$$$\int$$ t$$\vec{F}$$ x $$\frac{d\vec{r}}{dt}$$dt

3. The Attempt at a Solution
I set r = tR, where R = <x,y,z>. Taking the crossproduct F x R, I get <0,0,xy$$^{2}$$ + $$\frac{xy^{2}}{2}$$>. I then have an integral:

Int (t <0,0,xy$$^{2}$$ + $$\frac{xy^{2}}{2}$$>)dt which gives me
<0,0,xy$$^{2}$$ + $$\frac{xy^{2}}{2}$$>1/2

this simplifies to
<0,0,$$\frac{3xy^{2}}{4}$$>, which is an incorrect answer ( the right being <0,0,xy^2/2>.

I honestly have no clue what I'm doing wrong. My book doesn't show any examples, and doesn't explain how they get this formula so It's been a plug&chug problem for me that hasn't worked. I tried to look online for a description of what is going on but I havent found anything. If anyone knows where I could find examples to similar problems online and maybe where I could find how to derive the formula i used, or could help me out I would appreciate it greatly.