${\int }_{-1}^2{x}^{-2} \mathit{\ⅆ}x$

$\=\underset{s\to 0-}{lim}{\int }_{-1}^s{x}^{-2} \mathit{\ⅆ}x\+\underset{t\to 0\+}{lim}{\int }_t^2{x}^2 \mathit{\ⅆ}x$

$\=\underset{s\to 0-}{lim}\left({\left[-x^{-1}\right]}_{-1}^s\right)\+\underset{t\to 0\+}{lim}\left({\left[-x^{-1}\right]}_t^2\right)$

$\=\underset{s\to 0-}{lim}\left(-\frac{1}{s}\+1\right)\+\underset{t\to 0\+}{lim}\left(-\frac{1}{2}\+\frac{1}{t}\right)$

$\= \infty \+\infty$

$\= \infty$

Evaluate the integral

${\int }_{-1}^2{x}^{-2} \mathit{\ⅆ}x$ = $\mathrm{\∞}$$$

Evaluate ${\int }_{-1}^0{x}^{-2} \mathit{\ⅆ}x$ from first principles

${\int }_{-1}^s{x}^{-2} \mathit{\ⅆ}x$$$$\stackrel{\text{assuming real range}}{\to }$$-\frac{s\+1}{s}$$\stackrel{\text{limit}}{\to }$$\mathrm{\∞}$$$

Evaluate ${\int }_0^2{x}^{-2} \mathit{\ⅆ}x$ from first principles

${\int }_t^2{x}^{-2} \mathit{\ⅆ}x$$\stackrel{\text{assuming real range}}{\to }$$-\frac{1}{2}\frac{t-2}{t}$$\stackrel{\text{limit}}{\to }$$\mathrm{\∞}$