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?

  • 1
    $\begingroup$ $\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
  • 1
    $\begingroup$ 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

It might be simpler to just write $$\begin{align} S &=\int_0^{5\pi}|\sqrt{2}\sin{(x+\pi/4)}|\mathrm{d}x\\ &=5\int_0^{\pi}|\sqrt{2}\sin{(x+\pi/4)}|\mathrm{d}x\\ &=5\left(\int_0^{3\pi/4}\sqrt{2}\sin{(x+\pi/4)}\mathrm{d}x-\int_{3\pi/4}^{\pi}\sqrt{2}\sin{(x+\pi/4)}\mathrm{d}x\right)\\ &=\cdots\\ \end{align}$$

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.