= Solution
Set $A=10x^4-2xy^3$ and $B=-3x^2y^2$. Their mixed derivatives agree: $A_y=-6xy^2=B_x$. Both components are smooth on all of $\mathbb R^2$, a <simply connected space>, so the <line integral> is path independent.
Choose a horizontal segment from $(0,0)$ to $(2,0)$ followed by a vertical segment to $(2,1)$. On the first, $y=0$ and $dy=0$; on the second, $x=2$ and $dx=0$. Thus
$$
I=\int_0^2 10x^4\,dx+\int_0^1(-12y^2)\,dy=64-4=\boxed{60}.
$$
The selected path and its orientation are shown below. No contribution comes from a differential whose coordinate is held constant on a segment.
\Image[/past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2015-resit-path-independent-integral.png]
{title=Two straight segments used to evaluate the path-independent integral}
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