Answer
$L = \int_1^3 {\sqrt {1 + {e^{2x}}} dx} $
Work Step by Step
$$\eqalign{
& {\text{Let }}y = {e^x},{\text{ }}1 \leqslant x \leqslant 3 \cr
& {\text{Differentiating}} \cr
& \frac{{dy}}{{dx}} = \frac{d}{{dx}}\left[ {{e^x}} \right] \cr
& \frac{{dy}}{{dx}} = {e^x} \cr
& \cr
& {\text{Use the arc lenght formula }} \cr
& L = \int_a^b {\sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} dx} \cr
& {\text{Substituting the limits and the derivative of the function}} \cr
& L = \int_1^3 {\sqrt {1 + {{\left( {{e^x}} \right)}^2}} dx} \cr
& {\text{Simplifying}} \cr
& L = \int_1^3 {\sqrt {1 + {e^{2x}}} dx} \cr} $$
