A scientist models the rate of temperature change, dTdt\frac{dT}{dt}dtdT, of a new alloy using the equation
dTdt=9t3t2+k \frac{dT}{dt} = \frac{9t}{3t^2 + k} dtdT=3t2+k9twhere t≥0t \ge 0t≥0 is the time in seconds and kkk is a positive constant.
Find
∫9t3t2+k dt \int \frac{9t}{3t^2 + k} \, dt ∫3t2+k9tdtGiven that the change in temperature between t=0t = 0t=0 and t=2t = 2t=2 is exactly ln125\ln 125ln125 degrees, determine the value of kkk.
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.