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Let the $p$ be a period of the $f(x) $

$f(x) = |\sin x + \cos x|$

Find the $S = \int_0 ^{5p } f(x) dx$


Here is my attempt. The answer is $10 \sqrt2:$ enter image description here

But whenever I did the answer dosen't come at all.

Is the answer not correct, isn't it?

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    $\begingroup$ $\cos (x + \pi) \neq \cos(x) $ $\endgroup$ – Riquelme Aug 14 at 8:23
  • $\begingroup$ Hmm I will try it again $\endgroup$ – se-hyuck yang Aug 14 at 8:25
  • $\begingroup$ If $f$ is a periodic function with period $p$, then $\int_y^{y+mp} f(z)dz$ will be independent of choice of $y$ for any integer $m$, you can compute your integral over a interval more convenient, e.g. $(-\frac{\pi}{4},-\frac{\pi}{4}+5\pi)$. $\endgroup$ – achille hui Aug 14 at 8:26
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    $\begingroup$ Clearly $S=5\int_0^pf(x)\,dx$. And, by periodicity $\int_0^\pi f=\int_{-\pi/4}^{3\pi/4}f$, an interval where you can ignore the absolute values. $\endgroup$ – Jyrki Lahtonen Aug 14 at 8:27
  • $\begingroup$ @Riquelme But $|\cos(x+\pi)|=|\cos x|$. $\endgroup$ – Jyrki Lahtonen Aug 14 at 8:28
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It might be simpler to just write $$\begin{align} S &=\int_0^{5\pi}|\sqrt{2}\sin{(x+\pi/4)}|\mathrm{d}x\\ &=5\int_0^{\pi}|\sqrt{2}\sin{(x+\pi/4)}|\mathrm{d}x\\ &=5\left(\int_0^{3\pi/4}\sqrt{2}\sin{(x+\pi/4)}\mathrm{d}x-\int_{3\pi/4}^{\pi}\sqrt{2}\sin{(x+\pi/4)}\mathrm{d}x\right)\\ &=\cdots\\ \end{align}$$

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  • $\begingroup$ Terrific solution!! Thank you. $\endgroup$ – se-hyuck yang Aug 14 at 8:48

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