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.
480 exam-style questions on AQA A Level Maths 1.11 H: Integration, covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.