Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 4 - Integration - 4.9 Evaluating Definite Integrals By Substitution - Exercises Set 4.9: 23



Work Step by Step

In order to integrate this function, we have to use an «u» substitution. Choose an u, derivate it: $ u =x^3+9$ $ du=3x^2dx$ $ \dfrac{1}{3}du =x^2dx$ Change limits: $x_1 = -1$ $u_1= (-1)^3+9=8$ $x_2 = 1$ $u_2=(1)^3+9=10$ Then substitute: \[ \dfrac{1}{3} \int_{8}^{10} u^{-1/2} du \] Then integrate \[ \dfrac{1}{3} \bigg [ \dfrac{u^{1/2}}{1/2} \bigg ]_{8}^{10} \] Apply calculus 2nd theorem and reduce: \[ \dfrac{2}{3} (\sqrt{10}-\sqrt{8}) \]
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