Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$?

Question

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$? That is, is the integral $$\int_{S^{n-1}}\frac{1}{\sqrt{|x_1|}}d\sigma(x)$$ finite? Where $\sigma$ means the volume measure on the unit sphere $S^{n-1}$, which is the same as the $n-1$-dimensional Hausdorff measure on $S^{n-1}$.

Answer 1

Let $S^d_r$ denote the $d$-dimensional sphere of radius $r$, and let $x=(x_1,\bar x)$ with $\bar x=(x_2,\dots,x_n)$ and let $p>\in\mathbb{R}$. Then $$ \int_{S^{n-1}}\frac{1}{|x_1|^p}\,d\sigma(x)=\int_{-1}^1\frac{1}{|x_1|^p}\int_{S^{n-2}_{\sqrt{1-|x_1|^2}}}d\sigma(\bar x)\,dx_1=\omega_n\int_{-1}^1\frac{(1-|x_1|^2)^{(n-2)/2}}{|x_1|^p}\,dx_1, $$ where $\omega_n$ is the surface area of the unit $n-2$-dimensional sphere. The last integral is convergent if $p<1$.