Let $B(0,r)$ be a ball of radius $r$ with centre $0$ and $B(x_0,r)$ be a ball of radius $r$ with centre $x_0$. Let $w:\mathbb{R}^N\to\mathbb{R}$ be defined by $w(x)=|x|^{\alpha}$ for some $\alpha\in\mathbb{R}\setminus\{0\}$. Then how to calculate the following integral: $$ \int_{B(x_0,r)}w(x)\,dx $$
I am trying to use the change of variable by substituting $x-x_0=z$, then $z\in B(0,r)$. Therefore we obtain $$ \int_{B(x_0,r)}w(x)\,dx=\int_{B(x_0,r)}|x|^{\alpha}\,dx=\int_{B(0,r)}|z+x_0|^\alpha\,dz. $$ Can you help me from here how to proceed?
Thank you very much.