Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.4 Trigonometric Substitutions - 7.4 Exercises: 23

Answer

\[ = {\sin ^{ - 1}}\,\left( {\frac{x}{6}} \right) + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{dx}}{{\sqrt {36 - {x^2}} }}} \hfill \\ \hfill \\ the\,\,integral\,\,contains\,\,{a^2} - {x^2}\,\,so \hfill \\ \hfill \\ x = 6\sin \theta \,\,\,\,then\,\,\,dx = 6\cos \theta d\theta \hfill \\ and\,\,\,\sqrt {36 - {x^2}} = 6\cos \theta \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \int_{}^{} {\frac{{dx}}{{\sqrt {36 - {x^2}} }}} = \int_{}^{} {\frac{{6\cos \theta d\theta }}{{6\cos \theta }}} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \int_{}^{} {d\theta } \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \theta + C \hfill \\ \hfill \\ where\,\,\,\,\,\theta = {\sin ^{ - 1}}\,\left( {\frac{x}{6}} \right) \hfill \\ \hfill \\ then \hfill \\ \hfill \\ = {\sin ^{ - 1}}\,\left( {\frac{x}{6}} \right) + C \hfill \\ \hfill \\ \end{gathered} \]
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