Find, in simplest form,
∫3(2x−5)4 dx \int \frac{3}{(2x - 5)^4} \, \mathrm{d}x ∫(2x−5)43dxA population of bacteria grows at a rate R(t)=82t+1\displaystyle R(t) = \frac{8}{2t + 1}R(t)=2t+18 million per hour, where t t\,t is the time in hours, t≥0t \ge 0t≥0.
Show, by algebraic integration, that the total growth in the population between t=1t = 1t=1 and t=3t = 3t=3 is lnb \ln b\,lnb million, where b b\,b is a rational constant to be found.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.