# 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: $$\displaystyle \sum_{m=0}^{10} I_m = 55\pi$$.

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