The rate of mass accumulation in a sediment trap, MMM (in mg/year), is modeled by the function:
M(t)=4t3+5t2−12t+15t2+3,t≥0 M(t) = \frac{4t^3 + 5t^2 - 12t + 15}{t^2 + 3}, \quad t \ge 0 M(t)=t2+34t3+5t2−12t+15,t≥0where t t\,t is the time in years since the study began.
Given that
M(t)≡At+B+Ct+Dt2+3 M(t) \equiv At + B + \frac{Ct + D}{t^2 + 3} M(t)≡At+B+t2+3Ct+D(i) Find the values of the constants AAA, B B\,B and CCC.
(ii) Show that D=0D = 0D=0.
Hence, using algebraic integration, find the total mass accumulated between t=1t = 1t=1 and t=3t = 3t=3, giving your answer in the form p+qlnkp + q \ln kp+qlnk, where ppp, q q\,q and k k\,k are integers and k k\,k is prime.
