∫x^3 (√(1 − x)) dx


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Question Number 6343 by sanusihammed last updated on 24/Jun/16

$\int x^{3} \sqrt{1 - x} d x$
	Answered by nburiburu last updated on 24/Jun/16

	$b y s u b s t i t u t i o n t = \sqrt{1 - x} \Rightarrow x = 1 - t^{2}$ $d x = - 2 t d t$ $I = \int {(1 - t^{2})}^{3} . t . (- 2 t) d t$ $= \int (1 - 3 t^{2} + 3 t^{4} - t^{6}) (- 2 t^{2}) d t$ $= \int - 2 t^{2} + 6 t^{4} - 6 t^{6} + 2 t^{8} d t$ $= - \frac{2}{3} t^{3} + \frac{6}{5} t^{5} - \frac{6}{7} t^{7} + \frac{2}{9} t^{9} + c$ $a n d f i n a l l y$ $= - \frac{2}{3} (1 - x^{2}) . \sqrt{(1 - x^{2})} + \frac{6}{5} {(1 - x^{2})}^{2} . \sqrt{(1 - x^{2})} - \frac{6}{7} {(1 - x^{2})}^{3} . \sqrt{(1 - x^{2})} + \frac{2}{9} {(1 - x^{2})}^{4} . \sqrt{(1 - x^{2})} + c$
	Commented by sanusihammed last updated on 24/Jun/16

	$t h a n k s s o m u c h$
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