The rate of depletion of a mineral resource, R(t)R(t)R(t) in gigatonnes per decade, is modeled by the formula
R(t)=7t62−53t6+14,t≥1 R(t) = \frac{7t^6}{2} - \frac{5}{3t^6} + \frac{1}{4}, \quad t \ge 1 R(t)=27t6−3t65+41,t≥1where t t\,t is the number of decades since the start of industrial mining.
Determine the general expression for the total quantity of the resource depleted, ∫R(t) dt\int R(t) \, dt∫R(t)dt, giving each term in its simplest form.
