We have the following curve $\alpha(t) = (e^t\cos(t), e^t\sin(t))$. And I used the following formula to reparametrize the curve by arc length:
$$s(t) =\int_0^t \|\alpha'(\tau)\| \, d\tau.$$ Then I got $t = \ln\left(\dfrac{s+ \sqrt{2}}{2}\right).$
But according to our solutions we replace $t$ with $\ln(\frac{s}{\sqrt{2}})$. Is it possible to have more than one reparametrization?
Best Answer
Neither your answer nor the one that was given to you is correct. Note that\begin{align}s(t)&=\int_0^t\sqrt2e^t\,\mathrm dt\\&=\sqrt2\left(e^t-1\right).\end{align}So, you should take$$t=\log\left(1+\frac s{\sqrt2}\right).$$With this choice, $s=0\iff t=0$ as it should be, which fails to happen with any of those two solutions.