TITLE: Is $\frac{1}x\cos\left(\frac{1}x\right)$ Lebesgue integrable on $[0,1]$?
QUESTION [6 upvotes]: I have to decice if the following function is Lebesgue-integrable on $[0,1]$:
$$g(x)=\frac{1}x\cos\left(\frac{1}x\right) $$ where $x\in[0,1]$.

REPLY [2 votes]: $$
\int_{\large\frac1{n\pi}}^1\left|\frac1x\cos\left(\frac1x\right)\right|\,\mathrm{d}x
=\int_1^{n\pi}\left|\frac1x\cos(x)\right|\,\mathrm{d}x
$$
However, $\left[\frac\pi2,n\pi-\frac\pi2\right]\subset[1,n\pi]$ and
$$
\begin{align}
\int_{\pi/2}^{n\pi-\pi/2}\left|\frac1x\cos(x)\right|\,\mathrm{d}x
&=\sum_{k=1}^{n-1}\int_{k\pi-\pi/2}^{k\pi+\pi/2}\left|\frac1x\cos(x)\right|\,\mathrm{d}x\\
&\ge\sum_{k=1}^{n-1}\frac2{(2k+1)\pi}\left|\int_{k\pi-\pi/2}^{k\pi+\pi/2}\cos(x)\,\mathrm{d}x\right|\\
&=\sum_{k=1}^{n-1}\frac4{(2k+1)\pi}
\end{align}
$$
and since the Harmonic Series diverges, $\frac1x\cos\left(\frac1x\right)$ is not Lebesgue integrable on $[0,1]$.