# find the arclength parametrization of the Archimedean spiral?

How to find the arc length parametrization of the Archimedean spiral?

I know the curve is defined in the complex nubmer like this: $c(t)=e^{it} - ite^{it}$

and $|\frac{d}{dt}(c(t))|= te^{it}$ (is this correct?)

now I should find $\int te^{it} dt$ (the integration from 0 to t ,,,I am stuck here)

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It is easier to use the polar formula for the Archimedean spiral: $r=a+b\theta$. The arc length in polar coordinates is $ds=\sqrt{dr^2 +r^2 d\theta^2}$ or $\frac {ds}{d\theta}=\sqrt{r^2+\frac {dr^2}{d\theta^2}}=\sqrt{(a+b\theta)^2+b^2}$ and integrate that.