I would like to calculate the integral
$$\intop_{B(x,r)}\left(1+\left|y\right|\right)^{\alpha}\,dy,$$ where $x\in\mathbb{\mathbb{R}^{n}}$, $\alpha>0$ and $B(x,r)=\left\{ y\in\mathbb{R}^{n}:\:|x-y|<r\right\}$.
I tried substitution $y=\frac{z-x}{r}$ or using Tonelli theorem and considering the sphere, but I always had a problem with the absolute value. Do you have any other ideas or tips?
Hint: By using translation invariance of Lebesgue's measure and changing to polar coordinates you get
$$\int_{B(x,r)}\left(1+\left|y\right|\right)^{\alpha}\,dy= n V_n\int_0^r\left(1+s\right)^\alpha s^{n-1}\,ds,$$
where $V_n$ denotes the volume of the $n$-dimensional unit ball.