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+25e2t+1,0≤t≤4Using 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∫04R(t)dt, can be expressed in the form
∫abku2eu du \int_{a}^{b} k u^2 e^u \, du ∫abku2euduwhere aaa, b b\,b and k k\,k are constants to be found.
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.
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.
