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: 14


$-\frac{(4-y^3)^{5/3}}{5} + C$

Work Step by Step

Evaluate the Integral using substitution: $\int y^2(4-y^3)^{2/3}dy$ Substitution Rule: $\int f(g(x))gā€™(x)dx = \int f(u)du$ $u= 4-y^3$ $du =-3y^2$ Since $du$ in the expression is equal to $y^2$ it must be multiplied by $-\frac{1}{3}$ Solve the integral in terms of $u$: $\int (u)^{2/3}(-\frac{1}{3})du$ $-\frac{1}{3}\int u^{2/3}du $ $-\frac{1}{3} \frac{3u^{5/3}}{5} + C$ $-\frac{u^{5/3}}{5} + C$ Substitute for $u$: $-\frac{(4-y^3)^{5/3}}{5} + C$
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