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

Answer 1

The Correct Option is C

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