Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.5 Partial Fractions - 7.5 Exercises: 92

Answer

\[ = \ln \left| {1 + \tan \,\left( {\frac{x}{2}} \right)} \right| + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{dx}}{{1 + \sin x + \cos x}}} \hfill \\ \hfill \\ substitution \hfill \\ \hfill \\ set\,\,x = 2{\tan ^{ - 1}}\,\left( u \right)\,\,\,\,\,\,\,then\,\,\,\,\,\,\,\,du = 2\left( {\frac{{du}}{{1 + {u^2}}}} \right) \hfill \\ and\,\,\cos x = \frac{{1 - {u^2}}}{{1 + {u^2}}}\,\,\,then\,\,\,\,\sin x = \frac{{2u}}{{1 + {u^2}}} \hfill \\ \hfill \\ substitute\,\,for\,dx{\text{ }}\sin x{\text{ and }}\cos x \hfill \\ \hfill \\ \int_{}^{} {\frac{{dx}}{{1 + \sin x + \cos x}}} = \int_{}^{} {\frac{{2\frac{{du}}{{1 + {u^2}}}}}{{1 + \frac{{2u}}{{1 + {u^2}}} + \frac{{1 - {u^2}}}{{1 + {u^2}}}}}} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \int_{}^{} {\frac{{2du}}{{2 + 2u}}} \hfill \\ \hfill \\ integrating \hfill \\ \hfill \\ = \ln \left| {1 + u} \right| + C \hfill \\ \hfill \\ substitute\,\,for\,\,\,u \hfill \\ \hfill \\ = \ln \left| {1 + \tan \,\left( {\frac{x}{2}} \right)} \right| + C \hfill \\ \end{gathered} \]
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