## Calculus with Applications (10th Edition)

$$\frac{1}{5}$$
\eqalign{ & \int_{ - \infty }^{ - 5} {{x^{ - 2}}} dx \cr & {\text{by the definition of an improper integral}}{\text{}} \cr & \int_{ - \infty }^{ - 5} {{x^{ - 2}}} dx = \mathop {\lim }\limits_{a \to - \infty } \int_a^{ - 5} {{x^{ - 2}}} dx \cr & = \mathop {\lim }\limits_{a \to - \infty } \int_a^{ - 5} {{x^{ - 2}}} dx \cr & {\text{integrating by using the power rule}} \cr & = \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{{{x^{ - 2 + 1}}}}{{ - 2 + 1}}} \right)_a^{ - 5} \cr & = \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{{{x^{ - 1}}}}{{ - 1}}} \right)_a^{ - 5} \cr & = - \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{1}{x}} \right)_a^{ - 5} \cr & {\text{property of integrals}} \cr & = \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{1}{x}} \right)_{ - 5}^a \cr & {\text{use fundamental theorem of calculus }}\int_a^b {f\left( x \right)} dx = F\left( b \right) - F\left( a \right).\,\left( {{\text{see page 388}}} \right) \cr & = \mathop {\lim }\limits_{a \to - \infty } \left( {\frac{1}{a} - \frac{1}{{ - 5}}} \right) \cr & {\text{evaluate the limit when }}b \to \infty \cr & = \frac{1}{\infty } + \frac{1}{5} \cr & = \frac{1}{5} \cr}