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| Indefinite Integral https://forumdematematica.org/viewtopic.php?f=6&t=319 |
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| Autor: | kinu [ 23 abr 2012, 03:33 ] |
| Título da Pergunta: | Indefinite Integral |
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 \(\displaystyle \int_{0}^{100}f(x)dx\) |
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| Autor: | Rui Carpentier [ 25 abr 2012, 22:33 ] |
| Título da Pergunta: | Re: Indefinite Integral |
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 \(\int_{-8}^{16}f=0\) |
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