The rate of power consumption, P(t)P(t)P(t) in kilowatts, of an industrial cooling system is modeled by the function
P(t)=18t+45e2t+5 P(t) = \sqrt{18t+45} e^{\sqrt{2t+5}} P(t)=18t+45e2t+5where t≥0t \ge 0t≥0 is the time in hours since the start of the shift.
By using the substitution u=2t+5u = \sqrt{2t+5}u=2t+5, show that the total energy consumed in the interval 2≤t≤102 \le t \le 102≤t≤10, given by ∫210P(t) dt\int_{2}^{10} P(t) \, dt∫210P(t)dt, can be expressed in the form
∫abku2eu du \int_{a}^{b} k u^2 e^u \, du ∫abku2euduwhere aaa, bbb and kkk are integers to be found.
Hence find, by algebraic integration, the exact value of the total energy consumed between t=2t = 2t=2 and t=10t = 10t=10, giving your answer in the form pe5+qe3p e^5 + q e^3pe5+qe3 where ppp and qqq are integers.
