What is the length of a sine wave y = sin(x), from 0 to 2PI?

The standard way of computing it is to intergrate it from 0 to 2PI, as follows

y = sin(x)

Intergrate(sqrt(1 + (dy/dx)^2), 0, 2PI, dx)
(Yes, I'm creating the intergrate function on the fly, but you can see the logic, if you know calculus and parametric equations.)

This leads to the following:
Intergrate(sqrt(1 + (cos(x))^2), 0, 2PI, dx)
Now, I'm stuck. where can I go from here?

And don't make the mistake of saying
1 + (cos(x))^2 = sin(x)^2

The correct substitution for that is
1 - (cos(x))^2 = sin(x)^2

The one thing I can use (I guess) is symmetry.
I know the length of a cosine wave is the same as a sine wave. therefore
Len(sine wave) = Len(cos wave),
or
Intergrate(sqrt(1 + (cos(x))^2), 0, 2PI, dx) =
Intergrate(sqrt(1 + (sin(x))^2), 0, 2PI, dx)

where do I go from here?

If I can't directly solve it, how do I approximate it?

What value does it approach?

(just doing some recreational math)

Samwise Galenorn
[email protected]