Given $\alpha<0$ and $N\in\mathbb{N}$, let us consider the two integrals:
- $\displaystyle \int_{\mathbb{R}^N} e^{\alpha \|x\|^2}dx$
- $\frac{2|\alpha|}{N}\displaystyle \int_{\mathbb{R}^N} \|x\|^2e^{\alpha\|x\|^2}dx$.
I have proved that both of them are equal for $N=1,2,3$. But I don't know how to prove the equality in higher dimensions. Could someone help me?
Hint: The angular parts don't matter, could you somehow prove that
$$N\int_0^\infty r^{N-1}e^{\alpha r^2}\:dr = -2\alpha\int_0^\infty r^{N+1}e^{\alpha r^2}\:dr$$