Answer
$\frac{15}{4}$
Work Step by Step
Using arc length formulas.
\[
\begin{array}{c}
\int_{a}^{b} \sqrt{\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}+\left(\frac{d z}{d t}\right)^{2}}=L \\
\left(\frac{d x}{d t}\right)^{2}+\left(\frac{d y}{d t}\right)^{2}+\left(\frac{d z}{d t}\right)^{2}=\left(\sqrt{2} c^{\sqrt{2} t}\right)^{2}+\left(-\sqrt{2} c^{-\sqrt{2} t}\right)^{2}+4= \\
=8 \cosh ^{2}(\sqrt{2} t) \\
(\sqrt{8}=2 \sqrt{2}) \\
L=\int_{0}^{\sqrt{2} \ln 2} 2 \sqrt{2} \cosh (\sqrt{2} t) d t \\
=2 \sinh (\sqrt{2} t)]_{0}^{\sqrt{2} \ln 2} \\
=2 \sinh (2 \ln 2) \\
=\frac{15}{4}
\end{array}
\]
