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: 37

Answer

\[ = \frac{2}{3}\,{\left( {\tan x} \right)^{\frac{3}{2}}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\sec }^2}x{{\tan }^{\frac{1}{2}}}xdx} \hfill \\ \hfill \\ set\,\,\,u = \tan x\,\,\,\,\,then{\text{ }}du = \,\,{\sec ^2}xdx \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \int_{}^{} {{{\sec }^2}x{{\tan }^{\frac{1}{2}}}xdx} = \int_{}^{} {{u^{\frac{1}{2}}}du} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \frac{2}{3}{u^{\frac{3}{2}}} + C \hfill \\ \hfill \\ substituting\,\,back\,\,u\, = \,\tan x \hfill \\ \hfill \\ = \frac{2}{3}\,{\left( {\tan x} \right)^{\frac{3}{2}}} + C \hfill \\ \hfill \\ \end{gathered} \]
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