Integral bee: int{0,1/2} cos(1-cos(1-cos(...(1-cos(x))...) dx

[JAJAUSD123]

While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked.I would appreciate your support for this beautiful integral.
[math]\int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x[/math]

 

[blamocur]

A hint:
[math]f(x) = \cos (1- \cos (1-\cos(... (1-\cos(x))...) \longrightarrow f=\cos (1-f)[/math]

 

[JAJAUSD123]

A hint:
[math]f(x) = \cos (1- \cos (1-\cos(... (1-\cos(x))...) \longrightarrow f=\cos (1-f)[/math]

How could that help me to solve the problem?

 

[Dr.Peterson]

How could that help me to solve the problem?

Have you given any thought to what function your integrand is? Does it even converge? He's calling it [imath]f(x)[/imath] and considering the implications, in order to answer questions like that.

What do you get when you solve the equation? (If you don't see it, try graphing both sides.)

One thing you might do in thinking about that is to consider the sequence of functions given by [imath]f_{k+1}(x) = \cos(1-f_k(x))[/imath]. Graphing these gives interesting results. (I considered showing it to you, but decided you need to do some thinking of your own.)

 

[BigBeachBanana]

Integral $\int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x$

While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked. I would appreciate your support for this beautiful inte...

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