Find $\displaystyle \int_{1}^{\,2} {x^{-2}}\, dx.$

Hint: Choose $x_i^{\ast}$ to be the geometric mean of $x_{i-1}$ and $x_{i}$ (that is, $x_i^{\ast} = \sqrt{x_{i-1}x_i}$) and use the identity $$\frac{1}{m(m+1)} = \frac{1}{m} - \frac{1}{m+1}$$

Value of integral =