Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 8: Techniques of Integration - Section 8.2 - Integration by Parts - Exercises 8.2 - Page 455: 4

Answer

\begin{aligned} \int x^{2}\sin xd x =(2-x^2) \cos x+2x\sin x +C \end{aligned}

Work Step by Step

Given $$\int x^{2} \sin x d x $$ So, we have \begin{array}{|c c c|}\hline Differentiation && {Integration} \\ \hline x^{2} & + &\sin x \\ &\searrow&\\ \hline 2x &- &-\cos x \\ &\searrow&\\ \hline 2 & + &- \sin x \\ &\searrow&\\ \hline 0 & &\cos x \\ &&\\ \hline \end{array} Therefore \begin{aligned} I&=\int x^{2}\sin xd x\\ &=-x^2\cos x+2x\sin x+2\cos x+C\\ &=(2-x^2) \cos x+2x\sin x +C \end{aligned}
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