Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 7 - Integration - 7.2 Substitution - 7.2 Exercises: 21

Answer

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

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{2x + 1}}{{\,{{\left( {{x^2} + x} \right)}^3}}}dx} \hfill \\ Let\,\,u = {x^2} + x\,,\,\,So\,\,that \hfill \\ du = \,\left( {2x + 1} \right)dx \hfill \\ Then \hfill \\ \int_{}^{} {\frac{{2x + 1}}{{\,{{\left( {{x^2} + 1} \right)}^3}}}dx} = \int_{}^{} {\frac{{du}}{{{u^3}}}} \hfill \\ \int_{}^{} {{u^{ - 3}}du} \hfill \\ Power\,\,rule \hfill \\ \frac{{{u^{ - 3 + 1}}}}{{ - 3 + 1}} + C \hfill \\ - \frac{1}{{2{u^2}}} + C \hfill \\ Substituting\,\,u = {x^2} + x\,for\,\,u \hfill \\ - \frac{1}{{2\,{{\left( {{x^2} + x} \right)}^2}}} + C \hfill \\ \end{gathered} \]
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