The rate of biomass accumulation in a soil sample, BBB (measured in mg per day), is modeled as a function of the nitrogen concentration nnn (in parts per million) by the equation:
B(n)=5n−3n2−n,n>1 B(n) = \frac{5n - 3}{n^2 - n}, \quad n > 1 B(n)=n2−n5n−3,n>1Express B(n)B(n)B(n) in partial fractions.
Hence find ∫B(n) dn\displaystyle \int B(n) \, dn∫B(n)dn.
Use your answer to part (b) to find the value of kkk for which the total biomass accumulation between concentration levels n=kn = kn=k and n=2kn = 2kn=2k is exactly ln72\ln 72ln72, where k>1k > 1k>1.
∫k2kB(n) dn=ln72 \int_k^{2k} B(n) \, dn = \ln 72 ∫k2kB(n)dn=ln72438 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.