Answer
$$\frac{1}{2}\left(\sqrt{5}+\frac{1}{2}\ln \left(2+\sqrt{5}\right)\right)$$
Work Step by Step
\begin{aligned}
s &=\int_{a}^{b} \sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}} d t \\
&=\int_{0}^{1} \sqrt{ 1+4t^2} d t \\
&= t \sqrt{1+4t^{2}}+\frac{1}{2} \ln(2t+\sqrt{1+4t^{2}})\bigg|_{0}^{1}\\
&= \frac{1}{2}\left(\sqrt{5}+\frac{1}{2}\ln \left(2+\sqrt{5}\right)\right)
\end{aligned}
