A geologist models the rate of thermal energy emission, PPP, from a cooling volcanic vent using the equation
P(t)=8+2t3+lntt4,t≥1 P(t) = \frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1 P(t)=t48+2t3+lnt,t≥1where P P\,P is measured in megawatts and t t\,t is the time in hours since initial observation.
Find ∫lntt4 dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t∫t4lntdt
The total energy EEE, in megawatt-hours, emitted by the vent between t=1t = 1t=1 and t=2t = 2t=2 is given by the area under the curve P(t)P(t)P(t) between these two times.
Use your result from part (a) to calculate the exact value of EEE. Give your answer in the form a+lnba + \ln ba+lnb, where a a\,a and b b\,b are constants to be determined.
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.