prove that ∫_0 ^(2020) (x−[x])(√(x−[x])) dx=808


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Question Number 88177 by M±th+et£s last updated on 08/Apr/20

$p r o v e t h a t$ $\int_{0}^{2020} (x - [x]) \sqrt{x - [x]} d x = 808$
	Answered by TANMAY PANACEA. last updated on 08/Apr/20

	$\int_{0}^{1} (x - 0) \sqrt{x - 0} d x + \int_{1}^{2} (x - 1) (\sqrt{x - 1} d x + . . + \int_{2019}^{2020} (x - 2019) \sqrt{x - 2019} d x$ $=∣ \frac{x^{\frac{5}{2}}}{\frac{5}{2}} ∣_{0}^{1} + ∣ \frac{{(x - 1)}^{\frac{5}{2}}}{\frac{5}{2}} ∣_{1}^{2} + . . + ∣ \frac{{(x - 2019)}^{\frac{5}{2}}}{\frac{5}{2}} ∣_{2019}^{2020}$ $\frac{5 I}{2} = (1^{\frac{5}{2}} - 0^{\frac{5}{2}}) + (1^{\frac{5}{2}} - 0^{\frac{5}{2}}) + . . + (1^{\frac{5}{2}} - 0^{\frac{5}{2}})$ $= 2020$ $I = 2 \times 404 = 808$
	Commented by M±th+et£s last updated on 08/Apr/20

	$g o d b l e s s y o u s i r$
	Commented by TANMAY PANACEA. last updated on 08/Apr/20

	$m o s t w e l c o m e s i r . . .$
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