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Question Number 131852 by mnjuly1970 last updated on 09/Feb/21
$... analysis (II)... evaluate :: ∅=∫_1 ^( 10) x^2 d({x})=? {x} :: fractional part of x ...$
$. . . a n a l y s i s (I I) . . .$ $e v a l u a t e ::$ $\emptyset = \int_{1}^{10} x^{2} d ({x}) = ?$ ${x} :: f r a c t i o n a l p a r t o f x . . .$
	Answered by mr W last updated on 09/Feb/21
	$φ=∫_1 ^n x^2 d({x}) =Σ_(k=1) ^(n−1) ∫_k ^(k+1) x^2 d(x−k) =Σ_(k=1) ^(n−1) ∫_k ^(k+1) x^2 dx =(1/3)Σ_(k=1) ^(n−1) [(k+1)^3 −k^3 ] =(1/3)(n^3 −1) for n=10: φ=333$
	$ϕ = \int_{1}^{n} x^{2} d ({x})$ $= \underset{k = 1}{\sum^{n - 1}} \int_{k}^{k + 1} x^{2} d (x - k)$ $= \underset{k = 1}{\sum^{n - 1}} \int_{k}^{k + 1} x^{2} d x$ $= \frac{1}{3} \underset{k = 1}{\sum^{n - 1}} [{(k + 1)}^{3} - k^{3}]$ $= \frac{1}{3} (n^{3} - 1)$ $f o r n = 10 :$ $ϕ = 333$
	Commented by mr W last updated on 09/Feb/21

	$w h e r e i s t h e m i s t a k e s i r ?$
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