Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 52

Answer

\[ = \ln \left| {2 + \sqrt 3 } \right|\]

Work Step by Step

\[\begin{gathered} \int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\frac{{dy}}{{\sin y}}} \hfill \\ \hfill \\ use\,\,\,\frac{1}{{\sin y}} = \csc y \hfill \\ \hfill \\ then \hfill \\ \int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\frac{{dy}}{{\sin y}}} = \int_{\frac{\pi }{6}}^{\frac{\pi }{2}} {\csc ydy} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \left. { - \ln \left| {\csc \,y + \cot \,y} \right|} \right|_{\frac{\pi }{6}}^{\frac{\pi }{2}} \hfill \\ \hfill \\ evaluate\,\,the\,\,limits \hfill \\ \hfill \\ = \ln \left| {\csc \frac{\pi }{6} + \cot \frac{\pi }{6}} \right| - \ln \left| {\csc \frac{\pi }{2} + \cot \frac{\pi }{2}} \right| \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \ln \left| {2 + \sqrt 3 } \right| - \ln \left| {1 + 0} \right| \hfill \\ \hfill \\ = \ln \left| {2 + \sqrt 3 } \right| \hfill \\ \hfill \\ \end{gathered} \]
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