Evaluate $s(t)=\int^{t}_{-\infty}||\bf r\it'(u)||du$ for the Bernoulli spiral $\bf r\it(t)=\left<e^t\cos(3 t), e^t\sin(3 t)\right>$.
It is convenient to take $-\infty$ as the lower limit since $s(-\infty)=0$. Then use $s$ to obtain an arc length parametrization of $\bf r\it(t)$.

$\bf r\it_1(s)=\left< \right.$ , $\left. \right>$