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: 81

Answer

\[ = \frac{4}{3}\sqrt {1 + \sqrt x } \,\left( {\sqrt x - 2} \right) + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\frac{{dx}}{{\sqrt {1 + \sqrt x } }}} \hfill \\ \hfill \\ set\,\,\,x = \,{\left( {{u^2} - 1} \right)^2}\,\,\,\,\, \hfill \\ then\,\,\,dx = 4\,\left( {{u^2} - 1} \right)udu \hfill \\ \hfill \\ \int_{}^{} {\frac{{dx}}{{\sqrt {1 + \sqrt x } }}} \, = \int_{}^{} {\frac{{4\,\left( {{u^2} - 1} \right)udu}}{u}} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = 4\int_{}^{} {\,\left( {{u^2} - 1} \right)du} \hfill \\ \hfill \\ {\text{Integrate}} \hfill \\ \hfill \\ = 4\,\,\left[ {\frac{{{u^3}}}{3} - u} \right] + C \hfill \\ \hfill \\ substitute\,\,for\,\,\,u \hfill \\ \hfill \\ = \frac{4}{3}\sqrt {1 + \sqrt x } \,\left( {\sqrt x - 2} \right) + C \hfill \\ \hfill \\ \end{gathered} \]
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