Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises: 53

Answer

\[ = \frac{1}{2}\]

Work Step by Step

\[\begin{gathered} \int_1^3 {\frac{2}{{{x^2} + 2x + 1}}\,\,dx} \hfill \\ \hfill \\ factor\,ing\,\,the\,\,denominator \hfill \\ \hfill \\ = \int_1^3 {\frac{2}{{\,{{\left( {x + 1} \right)}^2}}}} \,\,dx \hfill \\ \hfill \\ set\,x + 1 = t\,\,\,\,\,\, \to \,\,\,\,\,dx = dt \hfill \\ \hfill \\ then \hfill \\ \hfill \\ = \int_2^4 {\frac{2}{{{t^2}}}\,dt} \, \hfill \\ \hfill \\ \,integrate\,and\,use\,\,the\,\,ftc \hfill \\ \hfill \\ = \,\,\left[ { - \frac{2}{t}} \right]_2^4 = - 2\,\,\left[ {\frac{1}{4} - \frac{1}{2}} \right] = \frac{1}{2} \hfill \\ \end{gathered} \]
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