Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 4 - Integration - 4.5 Exercises: 49

Answer

$\int$ $x^{2}\sqrt (1-x)$ $dx $ $=$ $\frac{-2}{3}$ $(1-x)^{3/2} $ $+$ $\frac{4}{5}$ $(1-x)^{5/2}$ $-$ $\frac{2}{7}$ $(1-x)^{7/2}$ $+$ $C $

Work Step by Step

$\int$ $x^{2}\sqrt (1-x)$ $dx $ $Let $ $U=1-x $ $, $ $dU=-dx $ $, $ $x=1-U $ $\int$ $x^{2}\sqrt (1-x)$ $dx $ $=$ $-\int(1-U)^{2}$ $U^{1/2}dU $ $=$ $-\int(1-2U+U^{2})$ $U^{1/2}dU$ $=$ $-\int(U^{1/2}-2U^{3/2}+U^{5/2})dU $ $=$ $-(\frac{U^{3/2}}{3/2}$ $-2$ $\frac{U^{5/2}}{5/2}$ $+$ $ \frac{U^{7/2}}{7/2}$ $) $ $+$ $C $ $=$ $\frac{-2}{3}$ $U^{3/2}$ $+$ $\frac{4}{5}$ $U^{5/2}$ $-$ $\frac{2}{7}$ $U^{7/2}$ $+$ $C $ $=$ $\frac{-2}{3}$ $(1-x)^{3/2} $ $+$ $\frac{4}{5}$ $(1-x)^{5/2}$ $-$ $\frac{2}{7}$ $(1-x)^{7/2}$ $+$ $C $
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