Local integrability of $g(x_1,x_2,x_3)=\frac{x_1}{{\left\lVert x\right\rVert}^3}$

Question

By using polar coordinates, is it simple to show that $f(x)=\frac{1}{{\left\lVert x\right\rVert}^\alpha}$ is locally integrable in $\mathbb{R}^n$, provided that $\alpha<n$.

Fix now $n=3$ and consider the function in $\mathbb{R} ^3$ $$g(x_1,x_2,x_3)=\frac{x_1}{{\left\lVert x\right\rVert}^3}. $$ I'm struggling trying to prove that $g\in L^{1}_{loc}(\mathbb{R}^3)$.

Any hint would be really appreciated.

Answer 1

Hint: $$\left |\frac{x_1}{|x|^3}\right|= \frac{|x_1|}{|x|^3} \le \frac{|x|}{|x|^3}=\frac{1}{|x|^2}.$$