The rate of growth of a specific bacteria population in a petri dish, G(t)G(t)G(t) in thousands of bacteria per hour, is modeled by the function
G(t)=122+2t+4 G(t) = \frac{12}{2 + \sqrt{2t + 4}} G(t)=2+2t+412for 0≤t≤160 \le t \le 160≤t≤16, where t t\,t is the time in hours since the start of the experiment. Using the substitution u=2+2t+4u = 2 + \sqrt{2t + 4}u=2+2t+4, find the exact total increase in the population over the 16-hour period by calculating the value of
∫016122+2t+4 dt \int_{0}^{16} \frac{12}{2 + \sqrt{2t + 4}} \, dt ∫0162+2t+412dtGive your answer in the form a+blnca + b \ln ca+blnc, where a,b, a, b,\,a,b, and c c\,c are integers.
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.