Answer
$\dfrac{10}{3}$
Work Step by Step
Since, $L=\int_{1}^{4}\sqrt{(\dfrac{dx}{dt})^2+(\dfrac{dy}{dt})^2} dt$
Thus, $L=\int_{1}^{4} \sqrt{1+(1/4)(1/x-2+x)} dx=\int_{1}^{4} (1/2)\sqrt{x^{-1/2}+x^{1/2}}dx$
or, $L=(1/2)[2x^{1/2}+(2/3)x^{3/2}]_{1}^{4}$
Thus, $L=(1/2)[2+\dfrac{14}{3}]=\dfrac{10}{3}$
