The velocity, v(t)v(t)v(t) in m s−1\text{m s}^{-1}m s−1, of a high-altitude research drone during a specific phase of its descent is modeled by the function
v(t)=3t2+14t+10t+4,t≥0 v(t) = \frac{3t^2 + 14t + 10}{t + 4}, \quad t \ge 0 v(t)=t+43t2+14t+10,t≥0where ttt is the time in seconds from the start of the observation.
Express v(t)v(t)v(t) in the form
At+B+Ct+4 At + B + \frac{C}{t + 4} At+B+t+4Cwhere AAA, BBB, and CCC are integers to be determined.
By using algebraic integration, determine the total distance traveled by the drone between t=0t = 0t=0 and t=8t = 8t=8. Give your answer in the form p+qln3p + q \ln 3p+qln3, where ppp and qqq are integers.
