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1 Pure Mathematics

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Question 545

A geologist models the rate of thermal energy emission, PPP, from a cooling volcanic vent using the equation

P(t)=8+2t3+ln⁡tt4,t≥1 P(t) = \frac{8 + 2t^3 + \ln t}{t^4}, \quad t \ge 1 P(t)=t48+2t3+lnt​,t≥1

where P P\,P is measured in megawatts and t t\,t is the time in hours since initial observation.

a.

Find ∫ln⁡tt4 dt\displaystyle \int \frac{\ln t}{t^4} \,\text{d}t∫t4lnt​dt

[3]
b.

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+ln⁡ba + \ln ba+lnb, where a a\,a and b b\,b are constants to be determined.

[5]
Markscheme

1 Pure Mathematics Questions

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