Answer
$\dfrac{\sqrt {14}}{12}(e^6-1)$
Work Step by Step
$I=\int_{0}^{1} (t) e^{(2t) \cdot (3t)} \sqrt{{{(\dfrac{dx}{dt})^{2}}+{(\dfrac{dy}{dt})^{2}}}}dt$
$=\int_{0}^{1} (t) e^{(2t) \cdot (3t)} \sqrt {14} dt$
$=\sqrt {14} \int_{0}^{1} (t) e^{6t^2} dt$
Plug $6t^2=a ; da=12t dt$
$=\sqrt {14} \int_{0}^{6} \dfrac{e^{a}}{12} da$
$=\dfrac{\sqrt {14}}{12}[e^a] _{0}^{6}$
$=\dfrac{\sqrt {14}}{12}(e^6-1)$
