Calculus 8th Edition

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

Chapter 4 - Integrals - 4.5 The Substitution Rule - 4.5 Exercises - Page 346: 23


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

Work Step by Step

Evaluate the Integral using substitution: $\int \frac{z^2}{\sqrt[3] {1+z^3}}dz$ Substitution Rule: $\int f(g(x))gā€™(x)dx = \int f(u)du$ $u= 1+z^3$ $du =3z^2$ Since $du$ in the expression is equal to $(z^2)$ it must be multiplied by $\frac{1}{3}$ Solve the integral in terms of $u$: $\int \frac{1}{\sqrt[3] u}(\frac13)du$ $\frac{1}{3}\int u^{-1/3}du $ $\frac{1}{3}\frac{3u^{2/3}}{2} +C$ $\frac{u^{2/3}}{2} + C$ Substitute for $u$: $\frac{(1+z^3)^{2/3}}{2} +C$
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