## Calculus with Applications (10th Edition)

Published by Pearson

# Chapter 8 - Further Techniques and Applications of Integration - 8.4 Improper Integrals - 8.4 Exercises - Page 452: 1

#### Answer

$$\frac{1}{3}$$

#### Work Step by Step

\eqalign{ & \int_3^\infty {\frac{1}{{{x^2}}}} dx \cr & {\text{solve the improper integral using the definition }}\int_a^\infty {f\left( x \right)} dx = \mathop {\lim }\limits_{b \to \infty } \int_a^b {f\left( x \right)} dx{\text{ }} \cr & {\text{then}} \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = \mathop {\lim }\limits_{b \to \infty } \int_3^b {\frac{1}{{{x^2}}}} dx{\text{ }} \cr & {\text{write }}\frac{1}{{{x^2}}}{\text{ as }}{x^{ - 2}} \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = \mathop {\lim }\limits_{b \to \infty } \int_3^b {{x^{ - 2}}} dx{\text{ }} \cr & {\text{integrate by using the power property }}\int {{x^n}} dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = \mathop {\lim }\limits_{b \to \infty } \left( {\frac{{{x^{ - 1}}}}{{ - 1}}} \right)_3^b \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = - \mathop {\lim }\limits_{b \to \infty } \left( {\frac{1}{x}} \right)_3^b \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 & \int_3^\infty {\frac{1}{{{x^2}}}} dx = - \mathop {\lim }\limits_{b \to \infty } \left( {\frac{1}{b} - \frac{1}{3}} \right) \cr & {\text{evaluate the limit when }}b \to \infty \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = - \left( {\frac{1}{\infty } - \frac{1}{3}} \right) \cr & {\text{Simplify}} \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = - \left( { - \frac{1}{3}} \right) \cr & \int_3^\infty {\frac{1}{{{x^2}}}} dx = \frac{1}{3} \cr}

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