Let $U \subset \mathbb{R}^n$ be a bounded open subset and $f(x) \in C_0^\infty(U)$. Why does the following equality hold for $1<k$, $0<l$ (assuming the integral is well-defined)? $$ \int_{U}\frac{f(x)^{k-1}}{x^l}x\cdot \nabla f dx=\frac{n-l}{k}\int_{U}\frac{f(x)^k}{x^l}dx $$

I think this is a standard vector calculus identity, but I don't see how things work. I relly appreciate your help.


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