Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - 3.9 Antiderivatives - 3.9 Exercises - Page 282: 11


$$F(x) =2 x^{3/2}-\frac{3}{2}x^{4/3}+C$$

Work Step by Step

Given $$f(x) = 3\sqrt{x}-2\sqrt[3]{x}= 3x^{1/2}-2x^{1/3}$$ Then by using if $f(x)=x^{\alpha}\ \to\ \ F(x) = \frac{x^{\alpha+1}}{\alpha+1}+C$ \begin{align*} F(x) &= \frac{ 3}{3/2}x^{3/2}-\frac{2}{4/3}x^{4/3}+C\\ &=2 x^{3/2}-\frac{3}{2}x^{4/3}+C \end{align*} To check \begin{align*} F'(x) &=3x^{\frac{1}{2}}-2x^{\frac{1}{3}}\\ &= 3\sqrt{x}-2\sqrt[3]{x}\\ &=f(x) \end{align*}
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