= Solution
Integrate the first component of $\mathbf F=\nabla\phi$ with respect to $x$:
$$
\phi=3xy^2-x^3yz-xz^2+g(y,z).
$$
Matching the $y$ component gives $g_y=0$, and matching the $z$ component gives $g_z=0$. Hence
$$
\boxed{\phi=3xy^2-x^3yz-xz^2+C.}
$$
The <fundamental theorem for line integrals> now yields
$$
\boxed{I=\phi(1,1,1)-\phi(0,0,0)=3-1-1=1.}
$$
The potential is a single-valued polynomial everywhere, so any piecewise smooth path with these endpoints has the same integral. The plus-gradient convention here should not be confused with the minus gradient of a mechanical potential energy.
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