Define a sequence(adsbygoogle = window.adsbygoogle || []).push({});

[tex]A_n(r) = \int_{-1}^1(1-x^2)^n \cos(rx)\, dx, \qquad n \in \mathbb{N}, r \in \mathbb{R}.

[/tex]

Prove that

[tex]A_n(r) = \frac{n!}{r^{2n+1}}[P_n(r)\sin r - Q_n(r)\cos r] [/tex]

where [tex]P_n[/tex] and [tex]Q_n[/tex] are two polynomials with integer coefficients. What is the degree of [tex]P_n[/tex] and of [tex]Q_n[/tex]?

Can anyone help me? Thanks.

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# How to prove that?

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