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.
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.