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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
