Calculus 8th Edition

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

Chapter 7 - Techniques of Integration - 7.4 Integration of Rational Functions by Partial Fractions - 7.4 Exercises - Page 541: 34

Answer

$\frac{1}{3}$ $x^{3}$$-$ $ln|x+1|$ $+$ $C$

Work Step by Step

Given that, $\frac {x^{5}+x-1}{x^{3}+1}$ Now, =$\frac {x^{5}+x-1}{x^{3}+1}$ = $x^{2}$$+$$\frac{-x^{2}+x-1}{x^{3}+1}$ =$x^{2}$$+$$\frac{-x^{2}+x-1}{(x+1)(x^{2}-x+1)}$ =$x^{2}$ $+$ $\frac{-1}{x+1}$ =$x^{2}$ $-$ $\frac{1}{x+1}$ So, $\int\frac {x^{5}+x-1}{x^{3}+1}dx$ =$\int(x^{2}-\frac {1}{x+1})dx$ =$\frac{1}{3}$ $x^{3}$$-$ $ln|x+1|$ $+$ $c$
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