Question

For $n\isin\N$ and $x\isin[1,\infin)$, let
$f_n(x)=\int\limits_{0}^{\pi}(x^2+(cos\theta)\sqrt{x^2-1})^nd\theta$
Then which one of the following is true?

• A. fn(x) is not a polynomial in x if n is odd and n ≥ 3.
• B. fn(x) is not a polynomial in x if n is even and n ≥ 4.
• C. fn(x) is a polynomial in x for all $n\isin\N$.
• D. fn(x) is not a polynomial in x for any n ≥ 3.

Answer 1

Answer: (C)

fn(x) is a polynomial in x for all $n\isin\N$.

Answer 2

The correct option is (C): fn(x) is a polynomial in x for all $n\isin\N$.