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.
306 exam-style questions on CCEA A Level Maths 1.1 Algebra and functions, covering 1.1.1 Algebra and functions, 1.1.2 Algebra and functions, 1.1.3 Algebra and functions, 1.1.4 Algebra and functions, 1.1.5 Algebra and functions, 1.1.6 Algebra and functions, 1.1.7 Algebra and functions, 1.1.8 Algebra and functions, 1.1.9 Algebra and functions, 1.1.10 Algebra and functions, 1.1.11 Algebra and functions, 1.1.12 Algebra and functions, 1.1.13 Algebra and functions, and 1.1.14 Algebra and functions. Each one has a worked solution and a mark scheme showing where the marks go.