Lebesgue Integrability of $\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$

Question

Given $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(0)=0$ and $f(x)=\left(\frac{1}{x}\right) \sin\left(\frac{1}{x}\right)$ for $x\in \mathbb{R}-\{0\}$, can someone please give me a rigorous proof explaining whether or not $f$ is Lebesgue integrable on $\mathbb{R}$? There is no problem at large values of $x$, but I am not able to write down clearly the behavior near the origin.

Answer 1

For positive integer $n$, using the change of variables $t = 1/x$, $$\int_{1/(\pi(n+1))}^{1/(\pi n)} |f(x)|\; dx = \int_{n\pi}^{(n+1)\pi} \dfrac{|\sin(t)|}{t}\; dt \ge \dfrac{1}{(n+1)\pi} \int_{n\pi}^{(n+1)\pi} |\sin(t)|\; dt = \dfrac{2}{(n+1)\pi}$$ Sum this over positive integers $n$.

Answer 2

The Lebesgue integrability of $\frac{1}{x}\sin\frac{1}{x}$ in a right neighbourhood of zero would imply the Lebesgue integrability of $\frac{\sin z}{z}$ over some interval of the form $(a>0,+\infty)$ by the change of variable $z=\frac{1}{x}$. However, $\frac{\sin x}{x}$ is not a Lebesgue integrable function over $(a>0,+\infty)$, despite it being an improperly Riemann integrable function over such set: $$ g(b)=\int_{a}^{b}\left|\frac{\sin x}{x}\right|\,dx $$ grows like $\frac{2}{\pi}\log\left(\frac{b}{a}\right)$, that is unbounded.