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: 36

Answer

\[ = - \frac{1}{{2\,\left( {{x^2} + 1} \right)}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{x}{{{x^4} + 2{x^2} + 1}}dx} \hfill \\ \hfill \\ factor\,\,the\,\,deno\min ator \hfill \\ \hfill \\ = \int_{}^{} {\frac{x}{{\,{{\left( {{x^2} + 1} \right)}^2}}}dx} \hfill \\ \hfill \\ set\,\,\,\left( {{x^2} + 1} \right) = t{\text{ then }}2xdx = dt\,\, \to \,\,xdx = \frac{{dt}}{2} \hfill \\ \hfill \\ integral\,becomes \hfill \\ \hfill \\ = \int_{}^{} {\frac{x}{{\,{{\left( {{x^2} + 1} \right)}^2}}}dx} = \int_{}^{} {\frac{{dt}}{{2{t^2}}}} \hfill \\ \hfill \\ {\text{Integrate using }}\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C{\text{ }}} \hfill \\ \hfill \\ = - \frac{1}{{2t}} + C \hfill \\ \hfill \\ substituting\,back\,t = \,\left( {{x^2} + 1} \right) \hfill \\ \hfill \\ = - \frac{1}{{2\,\left( {{x^2} + 1} \right)}} + C \hfill \\ \end{gathered} \]
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