Problem 57993. Easy Sequences 110: Integration of the Sum of a Recursive Trigonometric Function

A trigonometric function, , is defined as follows:
, x in radians
Applying recursively we define another function , for integer n:
We then define as the sum of value of R from 1 to n:
Finally, we are asked to evaluate the integral of S with respect to x, over the real range :
For example for , , , we have:
>> a = integral(@(x) sin(atan(x))+sin(atan(sin(atan(x))))+sin(atan(sin(atan(sin(atan(x)))))),pi,2*pi)
a = 7.05797686912156
Please present the final output rounded-off to 6 decimal places. Therefore the final answer is .
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NOTE: There are a number of ways to do numerical Integration in Matlab. Just make sure that the output would be accurate within 6 decimal places of the value obtained using the integral function shown above.
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Last Solution submitted on Jun 05, 2023

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