Let $\alpha:[-1,1]\rightarrow R^2$ be the curve segment given by $\alpha=(t,t^2)$.
If $\phi=v^2du+2uvdv$, (the fist component of $R^2$ is $u$ and the second one is $v$)
I have
$$\int_\alpha \phi=\int_{[-1,1]}\phi(\alpha'(t))dt=\int_{-1}^1(4t^2*0+2*1*2t*2)dt=4t^2|^1_{-1}=0$$
and also from $\phi=d(uv^2)$,
$$\int_\alpha \phi=\int_\alpha d(uv^2)=\int_{-1}^1d(t^3)=1+1=2$$
I'm pretty sure that one of them is wrong(or both are. hope not). Help me. Thanks.