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Question 660

A chemical bioreactor releases thermal energy at a rate R(t)R(t)R(t) MJ per hour, where t t\,t is the time in hours since the start of the reaction, given by

R(t)=50t+25e2t+1,0≤t≤4 R(t) = \sqrt{50t+25} e^{\sqrt{2t+1}}, \quad 0 \le t \le 4 R(t)=50t+25​e2t+1​,0≤t≤4
a.

Using the substitution u=2t+1u = \sqrt{2t+1}u=2t+1​, show that the total energy released, given by ∫04R(t) dt\int_{0}^{4} R(t) \, dt∫04​R(t)dt, can be expressed in the form

∫abku2eu du \int_{a}^{b} k u^2 e^u \, du ∫ab​ku2eudu

where aaa, b b\,b and k k\,k are constants to be found.

[5]
b.

Hence find, by algebraic integration, the exact value of the total energy released in the first 4 hours, giving your answer in the form Ae3+BeAe^3 + BeAe3+Be where A A\,A and B B\,B are integers.

[5]
Markscheme

Integration Questions

  1. A Level
  2. /Maths
  3. /Integration

864 exam-style questions on Edexcel A Level Maths Integration, 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. Each one has a worked solution and a mark scheme showing where the marks go.

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