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Calculus / Calculus: Early Transcendentals 1 / Chapter 7.7 / Problem 5E

Calculus: Early Transcendentals | 1st Edition | ISBN: 9780321570567 | Authors: William L. Briggs, Lyle Cochran, Bernard Gillett

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Infinite intervals of integration Evaluate the following

Chapter 7, Problem 5E

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QUESTION:

Infinite intervals of integration Evaluate the following integrals or state that they diverge.

$\int_{1}^{\infty} x^{-2} d x$

ANSWER:

Problem 5E

Infinite intervals of integration Evaluate the following integrals or state that they diverge. $\int\limits_{1}^{\infty{}}{x}^{-2}dx$

Solution

Step 1

In this problem we have to evaluate $\int\limits_{1}^{\infty{}}{x}^{-2}dx$ and then we have to state whether it diverges or not.

Consider $\int\limits_{1}^{\infty{}}{x}^{-2}dx$

By using $\int\limits_{a}^{\infty{}}f(x)dx=\lim\limits_{{b} \rightarrow {\infty{}}}\int\limits_{a}^{b}f(x)dx$ we get

$\int\limits_{1}^{\infty{}}{x}^{-2}dx=\lim\limits_{{b} \rightarrow {\infty{}}}\int\limits_{1}^{b}{x}^{-2}\ dx$

$=\lim\limits_{{b} \rightarrow {\infty{}}}\ [-\frac{1}{x}{{]}^{b}}_{1}$

Substituting the limits we get,

$=\lim\limits_{{b} \rightarrow {\infty{}}}\ [-{{\frac{1}{b}+1]}^{\ }}_{\ }$

Now considering $\lim\limits_{{b} \rightarrow {\infty{}}}$ we get

$\int\limits_{1}^{\infty{}}{x}^{-2}dx=1$

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Solution Found!

Infinite intervals of integration Evaluate the following

(choose chapter or problem)

Questions & Answers

Study Tools You Might Need

Chapter: 13 Problem: 51P

Chapter: 3 Problem: 16P

Chapter: 14 Problem: 9P

Chapter: 10 Problem: 1

Chapter: 0 Problem: 4

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