Calculus problem

Calculus Level 5

\[\large I_n =\int_{-\pi}^\pi \dfrac1{1 + 2^{\sin(x/2)}} \left( \dfrac{\sin(nx/2)}{\sin(x/2)} \right)^2 \, dx \]

For \(n=0,1,2,3,\ldots\), we define \(I_n\) as above.

How many of these choices are true?

Choice number 1: \(I_n = I_{n+1} \) for all \(n\leq 1 \).
Choice number 2: \(I_0, I_1, I_2, \ldots,I_n\) forms an arithmetic progression.
Choice number 3: \(\displaystyle \sum_{m=0}^9 I_{2m} = 90\pi \).
Choice number 4: \( \displaystyle \sum_{m=0}^{10} I_m = 55\pi \).

Submit your answer as the sum of values that represent the answer choice.

For example, if you think that choice number 1, 2, and 4 are true, submit your answer as \(1+2+4=7\).


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