Calculus with Applications (10th Edition)

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

Chapter 7 - Integration - 7.1 Antiderivatives - 7.1 Exercises: 23

Answer

\[ - \frac{{{\pi ^3}}}{{2{y^2}}} - 2\sqrt {\pi y} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\,\left( {\frac{{{\pi ^3}}}{{{y^3}}} - \frac{{\sqrt \pi }}{{\sqrt t }}} \right)} dy \hfill \\ Use\,\,the\,\,sum\,\,and\,\,difference\,\,rules \hfill \\ \int_{}^{} {\frac{{{\pi ^3}}}{{{y^3}}}dy} - \int_{}^{} {\frac{{\sqrt {11} }}{{\sqrt y }}} dy \hfill \\ Write\,\,\frac{1}{{{y^3}}}\,\,as\,\,{y^{ - 3}}\,\,and\,\frac{1}{{\sqrt y }}\,\, = \,\,{y^{ - 1/2}} \hfill \\ \int_{}^{} {{\pi ^3}{y^{ - 3}}dy} - \int_{}^{} {\sqrt \pi {y^{ - 1/2}}dy} \hfill \\ Use\,\,the\,\,power\,\,rule\,\,\int_{}^{} {{y^n}dn} = \frac{{{y^{n + 1}}}}{{n + 1}} + C \hfill \\ {\pi ^3}\,\left( {\frac{{{y^{ - 2}}}}{{ - 2}}} \right) - \sqrt \pi \,\left( {\frac{{{y^{1/2}}}}{{1/2}}} \right) + C \hfill \\ Simplifying \hfill \\ {\pi ^3}\,\left( {\frac{{{y^{ - 2}}}}{{ - 2}}} \right) - \sqrt \pi \,\left( {2{y^{1/2}}} \right) + C \hfill \\ - \frac{{{\pi ^3}}}{{2{y^2}}} - 2\sqrt {\pi y} + C \hfill \\ \hfill \\ \end{gathered} \]
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