Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.8 Improper Integrals - 7.8 Exercises: 27

Answer

\[ = \frac{1}{4}\]

Work Step by Step

\[\begin{gathered} \int_2^\infty {\frac{{dx}}{{\,{{\left( {x + 2} \right)}^2}}}} \, \hfill \\ \hfill \\ finding\,\,the\,\,anti\,derivative \hfill \\ \hfill \\ \int_{}^{} {\frac{{dx}}{{\,{{\left( {x + 2} \right)}^2}}}} \,\, = \,\, - \frac{1}{{x + 2}}\, + C \hfill \\ \hfill \\ use\,\,the\,\,Definition\,\,of\,\,improper\,\,integral\, \hfill \\ \hfill \\ \int_a^\infty {f\,\left( x \right)} \,dx = \,\,\,\,\,\mathop {\,\lim }\limits_{b \to \infty } \int_a^b {f\,\left( x \right)dx} \hfill \\ \hfill \\ \int_2^\infty {\frac{{dx}}{{\,{{\left( {x + 2} \right)}^2}}}} \,\,\, = \,\,\,\,\mathop {\,\lim }\limits_{b \to \infty } \,\int_2^b {\frac{{dx}}{{\,{{\left( {x + 2} \right)}^2}}}dx} \hfill \\ \hfill \\ = \,\,\,\mathop {\,\lim }\limits_{b \to \infty } \,\,\,\,\left[ { - \frac{1}{{x + 2}}} \right]_2^b \hfill \\ \hfill \\ use\,\,the\,\,ftc \hfill \\ \hfill \\ = \,\,\mathop {\,\lim }\limits_{b \to \infty } \,\,\,\,\left[ { - \frac{1}{{b + 2}} + \frac{1}{4}} \right] \hfill \\ \hfill \\ evaluate\,\,the\,\,\lim it \hfill \\ \hfill \\ = \frac{1}{4} \hfill \\ \end{gathered} \]
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