Answer
$\dfrac{10 \sqrt 5-2}{3}$
Work Step by Step
Here, we have
$ ds=\sqrt {(\dfrac{dx}{dt})^2+(\dfrac{dy}{dt})^2+(\dfrac{dz}{dt})^2} dt$
and $ds= \sqrt {1+x^2} dx$
This implies that
$L=\int_{0}^{2} (2x\sqrt {1+x^2} ) dx$
Plug $1+x^2 =p \implies 2x dx=dp$
or, $ \int_{1}^{5} \sqrt p dp= \dfrac{2}{3}[p^{3/2}]_{1}^{5}=\dfrac{10 \sqrt 5-2}{3}$
