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.
