Evaluate the line integral

$$\int \alpha\left(x^{2}+x y\right) d x+\beta\left(x^{2}+y^{2}\right) d y$$

with $\alpha$ and $\beta$ constants, along each of the following paths between the points $A=(1,0)$ and $B=(0,1)$ :

(i) the straight line between $A$ and $B$;

(ii) the $x$-axis from $A$ to the origin $(0,0)$ followed by the $y$-axis to $B$;

(iii) anti-clockwise from $A$ to $B$ around the circular path centred at the origin $(0,0)$.

You should obtain the same answer for the three paths when $\alpha=2 \beta$. Show that when $\alpha=2 \beta$, the integral takes the same value along any path between $A$ and $B$.