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 AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) 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.
