Answer
$${\text{Diverges}}$$
Work Step by Step
$$\eqalign{
& \int_{ - 1}^2 {\frac{3}{{{{\left( {x + 1} \right)}^2}}}} dx \cr
& {\text{The integrand }}\frac{3}{{{{\left( {x + 1} \right)}^2}}}{\text{ is not continuous at }}x = - 1,{\text{ then by }} \cr
& {\text{the definition of improper integrals}} \cr
& \int_{ - 1}^2 {\frac{3}{{{{\left( {x + 1} \right)}^2}}}} dx = \mathop {\lim }\limits_{r \to - {1^ + }} \int_r^2 {\frac{3}{{{{\left( {x + 1} \right)}^2}}}} dx \cr
& {\text{Integrating }} \cr
& = \mathop {\lim }\limits_{r \to - {1^ + }} \left[ { - \frac{3}{{x + 1}}} \right]_r^2 \cr
& = - \mathop {\lim }\limits_{r \to - {1^ + }} \left[ {\frac{3}{2} - \frac{3}{{r + 1}}} \right] \cr
& {\text{Evaluate the limit when }}r \to - {1^ + } \cr
& = - \left[ {\frac{3}{2} - \frac{3}{{ - {1^ + } + 1}}} \right] \cr
& = - \left[ {\frac{3}{2} - \frac{3}{{{0^ + }}}} \right] \cr
& = - \left[ {\frac{3}{2} - \infty } \right] \cr
& = \infty \cr
& {\text{Diverges}} \cr} $$
