Fórum de Matemática
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If \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a continuous function such that

\(\bullet f(x+4)+f(x-4) = f(x)\)

\(\bullet \displaystyle \int_{-8}^{16}f(x)dx = 12\) and \(\displaystyle \int_{0}^{4}f(x)dx = \frac{3}{4}\)

Then \(\displaystyle \int_{0}^{100}f(x)dx\)

Hint:

For any \(a\in \mathbb{R}\):

(1) \(\int_{a}^{a+4}f=\int_{a-4}^{a}f+\int_{a+4}^{a+8}f\). (Use \(f(x)=f(x-4)+f(x+4)\)).

(2) \(\int_{a}^{a+4}f +\int_{a+12}^{a+16}f=0\). (Use (1)).

(3) \(\int_{a}^{a+24}f=0\). (Use (2)).

(4) \(\int_{0}^{100}f=\int_{0}^{4}f\). (Use (3)).

We don't need to use the fact that \(\int_{-8}^{16}f=12\). Indeed, by (3), we should have \(\int_{-8}^{16}f=0\)