How long is a piece of sinosoidal string?

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I am trying to calculate the arc length of a sine wave.

Using s=\int_{}^{}\sqrt[]{1 + {(\frac{dy}{dx})}^{2}}dx

if y = sinx, dy/dx = cosx

So the integral simplyfies to s=\int_{}^{}\sqrt[]{1 + {cos}^{2}(x)}dx

However I do not know any integration technique (ie. substitution, by parts, etc..) with which I can calculate this integral analytically.

If you can think of any other way of going about this, any help would be greatly appreciated.

Benjy
 
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Mathematica calls this function EllipticE (performing the integral from 0 to 2п gives 4\sqrt{2}\mathtt{EllipticE}(1/2) \approx 7.6404), so I doubt there is a more elementary answer (like \pi/2 or \operatorname{arcsinh}(-1)).
 
Right, there is no elementary answer
 
I have to admit I am a little disappointed. I thought there might be a way of performing the integral by pure analytical means.
But thank you very much for your responses.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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