Solution
= Solution
A <line integral> uses $\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)$, including the tangent vector. With $\mathbf r(t)=(t,t,t)$, $0\leq t\leq1$, the three components sum to
$$
\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)=(4\alpha+2\alpha^2+\beta)t+\beta t^2.
$$
Consequently
$$
\int\mathbf F\cdot d\mathbf r
=\frac{4\alpha+2\alpha^2+\beta}{2}+\frac\beta3
=\boxed{\alpha^2+2\alpha+\frac56\beta}.
$$