Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises: 25

Answer

\[ = \frac{1}{{\cos \,t}} + \frac{{{{\tan }^2}t}}{2} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{\sin t + \tan t}}{{{{\cos }^2}t}}} dt \hfill \\ \hfill \\ split\,\;the{\text{ integrand}} \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\frac{{\sin t}}{{{{\cos }^2}t}} + \frac{{\tan \,t}}{{{{\cos }^2}t}}} \right)dt} \hfill \\ \hfill \\ for = \int_{}^{} {\frac{{\sin t}}{{{{\cos }^2}t}}dt} \hfill \\ \hfill \\ set\,\,\cos t = u\,\,\, \to \,\, - \sin t\,dt = du \hfill \\ \hfill \\ \int_{}^{} {\frac{{\sin t}}{{{{\cos }^2}t}}dt} = - \int_{}^{} {\frac{{dt}}{{{t^2}}} = \frac{1}{t} + {c_1} = \frac{1}{{\cos x}} + {c_1}} \hfill \\ \hfill \\ for = \int {\frac{{\tan \,t}}{{{{\cos }^2}t}} = } \int_{}^{} {\tan t{{\sec }^2}\,tdt} \hfill \\ \hfill \\ set\,\,\tan \,t = v\,\, \to \,\,{\sec ^2}tdt = dv \hfill \\ \hfill \\ \int {\frac{{\tan \,t}}{{{{\cos }^2}t}} = } \int_{}^{} {vdv} = \frac{{{v^2}}}{2} + {c_2} = \frac{{{{\tan }^2}t}}{2} + {c_2} \hfill \\ \hfill \\ {\text{Therefore}}{\text{,}} \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\frac{{\sin t}}{{{{\cos }^2}t}} + \frac{{\tan \,t}}{{{{\cos }^2}t}}} \right)dt} = \frac{1}{{\cos \,t}} + \frac{{{{\tan }^2}t}}{2} + C \hfill \\ \end{gathered} \]
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