Just a Cal 2 Problem

Calculus Level 3

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

Find the value of 11x2f(x)dx\int_{-1}^1 x^2f(x)\, dx

Bonus: Under the property f(x)+f(x)=kf(x)+f(-x)=k xR\forall x\in \mathbb{R} and some non-zero constant kk, find aax2nf(x)dx\int_{-a}^a x^{2n}f(x)\, dx where nZ+n\in \mathbb{Z}^+ and aR+a\in \mathbb{R}^+

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