\int _0^{\infty }\beta  x^2e^{-\alpha \frac{x^2}{2}}dx=\frac{1}{2}\int _{-\infty }^{\infty }\beta  x^2e^{-\alpha \frac{x^2}{2}}dx=\frac{\sqrt{2 \pi } \beta }{2\alpha ^{3/2}}=1