\[\begin{align*} E(X) &= \int_{x_m}^\infty x\cdot \frac{\alpha x_m^{\alpha}}{x^{\alpha+1}} dx \\ &= \alpha \int_{x_m}^\infty \frac{x_m^{\alpha}}{x^{\alpha}} dx \\ &= \alpha x_m^\alpha \int_{x_m}^\infty x^{-\alpha} dx \\ &= \alpha x_m^\alpha \left[\frac{x^{-\alpha+1}}{-\alpha+1} \right]_{x_m}^\infty, \alpha \ne 1 \\ &= \alpha x_m^\alpha \left[0 - \frac{x_m^{-\alpha+1}}{-\alpha + 1} \right] \\ &= \frac{\alpha}{\alpha - 1} \cdot \frac{x_m^\alpha}{x_m^{\alpha - 1}} \\ &= \frac{\alpha x_m}{\alpha - 1} \end{align*}\]
\[\begin{align*} E(X^2) &= \int_{x_m}^\infty x^2\cdot \frac{\alpha x_m^{\alpha}}{x^{\alpha+1}} dx \\ &= \alpha x_m^\alpha \int_{x_m}^\infty x^{-\alpha+1} dx \\ &= \alpha x_m^\alpha \left[\frac{x^{-\alpha+2}}{-\alpha+2}\right]_{x_m}^\infty \\ &= \alpha x_m^\alpha \left[0 - \frac{x_m^{-\alpha+2}}{-\alpha+2}\right] \\ &= \alpha x_m^\alpha \cdot \frac{x_m^{-\alpha+2}}{\alpha-2} \\ &= \frac{\alpha x_m^2}{\alpha-2} \\ \mathrm{Var}(X) &= \frac{\alpha x_m^2}{\alpha-2} - \left(\frac{\alpha x_m}{\alpha - 1} \right)^2 \\ &= \frac{\alpha(\alpha-1)^2 x_m^2 - \alpha^2 x_m^2(\alpha-2)}{(\alpha-1)^2(\alpha-2)} \\ &= \frac{(\alpha x_m^2)((\alpha-1)^2 - \alpha (\alpha-2))}{(\alpha-1)^2(\alpha-2)} \\ &= \frac{(\alpha x_m^2)(\alpha^2 - 2\alpha + 1 - \alpha^2 + 2\alpha)}{(\alpha-1)^2(\alpha-2)} \\ &= \frac{(\alpha x_m^2)}{(\alpha-1)^2(\alpha-2)} \end{align*}\]