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 $
