Let $$ K(x, y)=\frac{2 x_{n}}{n \alpha(n)} \frac{1}{|x-y|^n} $$ be the Poisson Kernel, where $x \in \mathbb{R}_{+}^n$ (the upper half-space of in $\mathbb{R}^n$ ), $y \in \mathbb{R}^n$, and $\alpha(n)$ is the volume of the $n$-dimensional unit ball. How do you show that $$ \int_{\partial \mathbb{R}_{+}^{n}} K(x, y) \, d y=1 \text{?} $$
There was an answer provided by Evaluating the Poisson Kernel in the upper half space in $n$-dimensions
I just didn't understand why the following step is correct. What is the justification? \begin{equation} \int_{\mathbb{R}^{n-1}} \frac{2 x_{n}}{n \alpha(n)} \frac{1}{|x-y|^n} d y=\frac{2 x_{n}}{n \alpha(n)} \int_{\mathbb{R}^{n-1}} \frac{1}{\left(x_n^2+y_1^2+\ldots+y_{n-1}^2\right)^{\frac{n}{2}}} \, d y \end{equation}