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: 33


$$\frac{2}{9} (1+x^3)^{3/2}+C$$

Work Step by Step

Given $$\int \sqrt{x^{4}+x^{7}} d x$$ Since \begin{align*} \int \sqrt{x^{4}+x^{7}} d x&=\int \sqrt{x^{4}(1+x^{3})} d x\\ &=\int x^2\sqrt{1+x^3}dx \end{align*} Let $$u=1+x^3,\ \ \ \ du=3x^2dx$$ Then \begin{align*} \int \sqrt{x^{4}+x^{7}} d x&=\int \sqrt{x^{4}(1+x^{3})} d x\\ &=\int x^2\sqrt{1+x^3}dx\\ &=\frac{1}{3} \int u^{1/2}du\\ &=\frac{2}{9} u^{3/2}+C\\ &=\frac{2}{9} (1+x^3)^{3/2}+C \end{align*}
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