For n ∈ N and x ∈ [1, ∞), let \(f_{n}(x)=\int_{0}^{\pi}\left(x^{2}+(\cos \theta) \sqrt{x^{2}-1}\right)^{n} d \theta\). Then which one of the following is true?
1
fn(x) is not a polynomial in x if n is odd and n ≥ 3.
2
fn(x) is not a polynomial in x if n is even and n ≥ 4.
3
fn(x) is a polynomial in x for all n ∈ N.
4
fn(x) is not a polynomial in x for any n ≥ 3.