In a biochemical model, the rate of increase of a substance's concentration, CCC, with respect to time, ttt, is given by dCdt=12t22t3+A\displaystyle \frac{dC}{dt} = \frac{12t^2}{2t^3 + A}dtdC=2t3+A12t2, where A A\,A is a positive constant.
Find
∫12t22t3+A dt \int \frac{12t^2}{2t^3 + A} \, dt ∫2t3+A12t2dtGiven also that
∫1312t22t3+A dt=ln196 \int_{1}^{3} \frac{12t^2}{2t^3 + A} \, dt = \ln 196 ∫132t3+A12t2dt=ln196find the value of AAA.
