Let $$K(x,y) = \frac{2x_n}{n \alpha(n)} \frac{1}{|x-y|^n}$$ 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 $$\int_{\partial \mathbb{R}_+^n} K(x,y) dy =1?$$
I tried doing this in simple cases (e.g. two dimensions), and it can out pretty cleanly (I think you can also probably use complex analysis if we're in two-dimensions?). However, I couldn't figure how to solve it in general $n$ dimensions, because the exponent to the $n$-th power was giving me trouble. How would one go about showing the general case?