The rate of production of a biofuel in a research facility, R(t)R(t)R(t) kilolitres per hour, is modelled by the formula
R(t)=4t3+5t2+8t+15t2+3,1≤t≤3 R(t) = \frac{4t^3 + 5t^2 + 8t + 15}{t^2 + 3}, \quad 1 \le t \le 3 R(t)=t2+34t3+5t2+8t+15,1≤t≤3where ttt is the time in hours from the start of a production cycle. Given that
R(t)≡At+B+Ct+Dt2+3 R(t) \equiv At + B + \frac{Ct + D}{t^2 + 3} R(t)≡At+B+t2+3Ct+D(i) find the values of the constants AAA, BBB and CCC.
(ii) show that D=0D = 0D=0.
Hence, using algebraic integration, calculate the total volume of biofuel produced between t=1t = 1t=1 and t=3t = 3t=3, giving your answer in the form p+qlnkp + q \ln kp+qlnk, where ppp, qqq and kkk are integers and kkk is prime.
