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: 24


$$\frac{2}{3}\tan^{-1}(x^{3/2})+C $$

Work Step by Step

Given $$\int \frac{\sqrt{x}}{x^{3}+1} d x$$ Let $$u=\sqrt{x},\ \ \ \ du=\frac{1}{2\sqrt{x}}dx $$ Then \begin{aligned} \int \frac{\sqrt{x}}{x^{3}+1} d x=& \int \frac{u}{u^{6}+1} 2 u \, d u \\ &=\int \frac{2 u^{2}}{u^{6}+1} d u \\ &=\int \frac{2 u^{2}}{\left(u^{3}\right)^{2}+1}\\ &= \frac{2}{3}\int \frac{3 u^{2}}{\left(u^{3}\right)^{2}+1}\\ &=\frac{2}{3}\tan^{-1}(u^3)+C\\ &=\frac{2}{3}\tan^{-1}(x^{3/2})+C \end{aligned}
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