formula

$EX_1&=&-\int_0^{\infty}r\log r\exp(-r^2/2)dr\\ &\geq &-\int_0^{\infty}r(r-1)\exp(-r^2/2)dr\\ &=&-\int_0^{\infty}r^2\exp(-r^2/2)dr+\int_0^{\infty} r\exp(-r^2/2)dr\\ &=&-\frac{1}{2}\int_0^{\infty}\sqrt{s}\exp(-s/2)ds+1\quad (s=r^2)\\ &=&-\frac{\Gamma (3/2)}{2^{3/2}}+1\\ &=&-\frac{\sqrt{\pi}}{4\sqrt{2}}+1>0$