Question #73487

Let $\vec{a}(t), \vec{b}(t), \vec{c}(t)$ be three non-coplanar differentiable vector functions of $t \in \mathbb{R}$ such that their scalar triple product $V(t) = [\vec{a}(t) \vec{b}(t) \vec{c}(t)] \ge 1$ for all $t \ge 0$, and $V(0) = 1$. Let $\vec{a}', \vec{b}', \vec{c}'$ be the reciprocal system of vectors corresponding to $\vec{a}, \vec{b}, \vec{c}$, and let $W(t) = [\vec{a}'(t) \vec{b}'(t) \vec{c}'(t)]$. Define the functions: $f(t) = \frac{d\vec{a}}{dt} \cdot \vec{a}' + \frac{d\vec{b}}{dt} \cdot \vec{b}' + \frac{d\vec{c}}{dt} \cdot \vec{c}'$ $g(t) = \left[ \frac{d\vec{a}'}{dt} \vec{b}' \vec{c}' \right] + \left[ \vec{a}' \frac{d\vec{b}'}{dt} \vec{c}' \right] + \left[ \vec{a}' \vec{b}' \frac{d\vec{c}'}{dt} \right]$ If $f(t)$ and $g(t)$ satisfy the relation $f(t) + g(t) = \frac{2t^3}{t^2+1} W(t)$ for all $t \ge 0$, find the value of the integral $\int_0^2 \left( f(t) + t W(t) \right) dt$.