Chapter 4 - Integration - 4.7 Rectilinear Motion Revisited Using Integration - Exercises Set 4.7 - Page 329: 5

$$\eqalign{ & \left( {\text{a}} \right)s\left( t \right) = {t^3} - {t^2} + 1 \cr & \left( {\text{b}} \right)s\left( t \right) = - \frac{1}{3}\sin 3t + 4t + 3 \cr} $$

$$\eqalign{ & \left( {\text{a}} \right){\text{ }}v\left( t \right) = 3{t^2} - 2t;{\text{ }}s\left( 0 \right) = 1 \cr & s\left( t \right) = \int {v\left( t \right)} dt \cr & s\left( t \right) = \int {\left( {3{t^2} - 2t} \right)} dt \cr & s\left( t \right) = {t^3} - {t^2} + C \cr & {\text{Use the initial condition }}s\left( 0 \right) = 1 \cr & 1 = {\left( 0 \right)^3} - {\left( 0 \right)^2} + C \cr & C = 1 \cr & {\text{Then,}} \cr & s\left( t \right) = {t^3} - {t^2} + 1 \cr & \cr & \left( {\text{b}} \right){\text{ }}a\left( t \right) = 3\sin 3t;{\text{ }}v\left( 0 \right) = 3;{\text{ }}s\left( 0 \right) = 3; \cr & v\left( t \right) = \int {a\left( t \right)} dt \cr & v\left( t \right) = \int {\left( {3\sin 3t} \right)} dt \cr & v\left( t \right) = - \cos 3t + C \cr & {\text{Use the initial condition }}v\left( 0 \right) = 3 \cr & 3 = - \cos 3\left( 0 \right) + C \cr & C = 4 \cr & {\text{Then,}} \cr & v\left( t \right) = - \cos 3t + 4 \cr & s\left( t \right) = \int {v\left( t \right)} dt \cr & s\left( t \right) = \int {\left( { - \cos 3t + 4} \right)} dt \cr & s\left( t \right) = - \frac{1}{3}\sin 3t + 4t + C \cr & {\text{Use the initial condition }}s\left( 0 \right) = 3 \cr & 3 = - \frac{1}{3}\sin 3\left( 0 \right) + 4\left( 0 \right) + C \cr & C = 3 \cr & {\text{Then,}} \cr & s\left( t \right) = - \frac{1}{3}\sin 3t + 4t + 3 \cr} $$