Calculus 8th Edition

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

Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 517: 55


$$x(\ln x)^{3}-3 x(\ln x)^{2}+6 x \ln x-6 x+C$$

Work Step by Step

Given $$ \int(\ln x)^{3} d x$$ by using the form $$ \int(\ln x)^{n} d x =x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$ \begin{align*} \int(\ln x)^{3} d x &=x(\ln x)^{3}-3 \int(\ln x)^{2} d x\\ &=x(\ln x)^{3}-3\left[x(\ln x)^{2}-2 \int(\ln x)^{1} d x\right] \\ &=x(\ln x)^{3}-3 x(\ln x)^{2}+6\left[x(\ln x)^{1}-1 \int(\ln x)^{0} d x\right] \\ &=x(\ln x)^{3}-3 x(\ln x)^{2}+6 x \ln x-6 \int 1 d x\\ &=x(\ln x)^{3}-3 x(\ln x)^{2}+6 x \ln x-6 x+C \end{align*}
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