Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 7 - Section 7.6 - Integration Using Tables and Computer Algebra Systems - 7.6 Exercises - Page 513: 31

Answer

$\displaystyle \frac{1}{5}\ln|x^{5}+\sqrt{x^{10}-2}|+C$

Work Step by Step

With $\left[\begin{array}{ll} u=x^{5} & \\ du=5x^{4}, & x^{4}dx=\dfrac{du}{5} \end{array}\right]$ the integral becomes $I=\displaystyle \frac{1}{5}\int\frac{du}{\sqrt{u^{2}-2}}.$ Table of integrals: $\color{blue}{43. \displaystyle \quad\int\frac{du}{\sqrt{u^{2}-\mathrm{a}^{2}}}=\ln|u+\sqrt{u^{2}-a^{2}}|+C }$ $(\mathrm{a}^{2}=2)$ $I=\displaystyle \frac{1}{5}\ln|u+\sqrt{u^{2}-2}|+C$ ... bring back $x$... $=\displaystyle \frac{1}{5}\ln|x^{5}+\sqrt{x^{10}-2}|+C$
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