Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 42


$$\frac{4}{5}\left(\sqrt{1+\sqrt{x}}\right)^{5}- \frac{4}{3}\left(\sqrt{1+\sqrt{x}}\right)^{3}+C$$

Work Step by Step

Given $$\int \sqrt{1+\sqrt{x}} d x$$ Let \begin{aligned} u &=\sqrt{1+\sqrt{x}} \\ u^{2} &=1+\sqrt{x} \\ \sqrt{x} &=u^{2}-1 \end{aligned} Then $ d x=4 u\left(u^{2}-1\right) d u$, and hence \begin{aligned} \int \sqrt{1+\sqrt{x}} d x &=4 \int u^{2}\left(u^{2}-1\right) d u \\ &=4 \int\left(u^{4}-u^{2}\right) d u\\ &= 4 \frac{u^{5}}{5}-4 \frac{u^{3}}{3}+C\\ &= \frac{4}{5}\left(\sqrt{1+\sqrt{x}}\right)^{5}- \frac{4}{3}\left(\sqrt{1+\sqrt{x}}\right)^{3}+C \end{aligned}
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