How does one compute the integral of $|x|^{- \alpha}$ over $B_1 (0) \subset R^n$, where $0 < \alpha < n$? I know the function is Lebesgue integrable because $\alpha < n$, but I'm need of an exact formula.
Integral of a function over the unit ball
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real-analysis
integration
multivariable-calculus
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0use spherical coordinates: $$I=\int_{||x||=1}||x||^{-a}d^nx=Vol(S^n)\int_0^1r^{n-a-1}dr=\frac{Vol(S^n)}{n-a}$$ where the volume of the unit sphere might be found here: https://en.wikipedia.org/wiki/N-sphere#Volume_and_surface_area – 2017-01-19