Answer
$$\frac{1}{4}$$
Work Step by Step
$$\eqalign{
& \int_1^\infty {\frac{1}{{{x^5}}}dx} = \int_1^\infty {{x^{ - 5}}dx} \cr
& {\text{Using the definition of improper integrals see page 1053}} \cr
& \underbrace {\int_a^{ + \infty } {f\left( x \right)dx = \mathop {\lim }\limits_{M \to + \infty } } \int_a^M {f\left( x \right)} dx}_ \Downarrow \cr
& \int_1^\infty {{x^{ - 5}}dx} = \mathop {\lim }\limits_{M \to \infty } \int_1^M {{x^{ - 5}}dx} \cr
& {\text{Integrating}} \cr
& = \mathop {\lim }\limits_{M \to \infty } \left[ {\frac{{{x^{ - 4}}}}{{ - 4}}} \right]_1^M \cr
& = - \frac{1}{4}\mathop {\lim }\limits_{M \to \infty } \left[ {\frac{1}{{{x^4}}}} \right]_1^M \cr
& = - \frac{1}{4}\mathop {\lim }\limits_{M \to \infty } \left[ {\frac{1}{{x^4 }}} \right]_1^M \cr
& = - \frac{1}{4}\mathop {\lim }\limits_{M \to \infty } \left[ {\frac{1}{{ M^4 }} - \frac{1}{{1^4 }}} \right] \cr
& = - \frac{1}{4}\mathop {\lim }\limits_{M \to \infty } \left[ {\frac{1}{{M^4 }} - 1} \right] \cr
& {\text{Evaluate the limit when }}M \to \infty \cr
& = - \frac{1}{4}\left[ {0 - 1} \right] \cr
& = \frac{1}{4} \cr} $$
