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How to calculate arc-length reparameterization of a regular curve.

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I am self-studying differential geometry, so please forgive my question if it is lacking in some way. I have a regular curve $\gamma : (0,\pi) \to \mathbb{R}^2$ defined by $$\gamma(t) = \left ( \sin (t), \cos(t) + \log \tan \frac{t}{2}\ \right )$$I know this is regular on the interval $(0,\pi/2)$ because the derivative$$\gamma'(t) = \left ( \cos(t), -\sin(t) + \csc(t) \right )$$is non-zero on that interval. How do I find$$\begin{split}\phi(t) &= \int\sqrt{ \cos^2(t)+ \left ( -\sin(t) + \csc(t) \right )^2} dt \\&=\int\sqrt{ \cos^2(t)+ \sin^2(t) -2\sin(t)\csc(t)+\csc^2(t)} \, \, dt \\&= \int \cot(t) \, dt = \ln(|\sin(t)|)+C\end{split} $$But how am I to evaluate this for some $t$ on the interval $(0,\pi/2)$?


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