Answer
$\dfrac{107\sqrt {14}}{12}$
Work Step by Step
$I=\int_{0}^{1} (1+t)^2 (2+3 t) \sqrt {{(\dfrac{dx}{dt})^{2}}+{(\dfrac{dy}{dt})^{2}}}dt$
$=\int_{0}^{1} (1+t)^2 (2+3 t) \sqrt {14}dt$
$=\sqrt {14} \int_{0}^{1} 3t^3+8t^2+7t+2 dt$
$=\sqrt {14}[\dfrac{3t^4}{4}+\dfrac{8t^3}{3}+\dfrac{7t^2}{2}+2t] _{0}^{1}$
$=\sqrt {14}[\dfrac{3(1)^4}{4}+\dfrac{8(1)^3}{3}+\dfrac{7(1)^2}{2}+2(1)] _{0}^{1}$
$=\dfrac{107\sqrt {14}}{12}$
