Just a Cal 2 Problem

Calculus Level 3

Let \(f:\mathbb{R} \to \mathbb{R}\) be a function with derivative of any order. Suppose \(f\) has the property that \(f(x)+f(-x)=1\) \(\forall x\in \mathbb{R}\)

Find the value of \[\int_{-1}^1 x^2f(x)\, dx\]

Bonus: Under the property \(f(x)+f(-x)=k\) \(\forall x\in \mathbb{R}\) and some non-zero constant \(k\), find \[\int_{-a}^a x^{2n}f(x)\, dx\] where \(n\in \mathbb{Z}^+\) and \(a\in \mathbb{R}^+\)

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