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**tttcomrader** Construct the involute for $\displaystyle y = \frac {x^2}{2} $ starting from (0,0).

My solution so far:

Let $\displaystyle \alpha (t) = (t, \frac {t^2}{2}) $

I use the formula $\displaystyle \alpha (t) - t(t) s(t) $ where $\displaystyle s(t) = \int ^{t} _{t_{0}} V(u)du $

I find that $\displaystyle t(t) = \frac { \alpha ^{'} } { | \alpha ^{'} | } = \frac { (1,t)} { \sqrt {1+t^2} } $

But in solving s(t), I have: $\displaystyle s(t) = \int ^{t} _{0} \sqrt {1+t^2}dt = \frac {t}{2} \sqrt {1+t^2} + \frac {1}{2} ln(t+ \sqrt {1+t^2} ) $, but I can't see a way to simple this, am I doing this right?

Thank you.