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I was doing the question $$\int_{-2}^{1} \{x\} dx =\frac{3}{2}$$ which got me wondering for the general expression and after few trials i got to the expression for $$\int_{a}^{b} \{nx\} dx = \left(\frac{[nb]-[na]}{2n}\right) + \left(\frac{\{(nb)^2\}-\{(na)^2\}}{2}\right) {n}\neq{0}$$ using graph of the function and Reimann summation can me making the proof a little more formal the derivation goes as: if $\{nb\}=0$ $$\int_{0}^{b} \{nx\} dx = \left(\sum_{x=0}^{\frac{1}{n}}nx +\sum_{\frac{1}{n}}^{\frac{2}{n}} ..... \sum_{x=\frac{nb-1}{n}}^{b} \right) \Delta{x} $$ and $$\sum_{\frac{k}{n}}^{\frac{k+1}{n}} {nx} - [nx]=0.5 $$ then $$\int_{0}^{b} \{nx\} dx = \frac{b}{2}$$ and if $\{nb\}\neq{0}$ $$\int_{0}^{b} = \int_{0}^{\frac{[nb]}{n}} \{nx\} dx + \int_{\frac{[nb]}{n}}^{b} \{nx\} = \frac{[nb]}{2n} + \frac{(\{nb\})^2}{2} $$ now using the properties of definite integral we can use the formula given above any

and i checked for ${a},{b}$ and ${n}$ as integers where the results hold and but i don't know for non-integers number

AlgTop1854
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Neelamber Mishra
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    What is your question? – user170231 Oct 17 '23 at 22:13
  • i have a expression but i don't know if it is correct or not . so,i tried to derive it but i don't know whether the derivation is correct or not . so i wanted help in finding a proof or if the the expression is incorrect it would be better if someone can write the correct epression – Neelamber Mishra Oct 18 '23 at 15:43
  • Okay, that should have been stated somewhere in the original question body. See How to ask a good question for more tips on how you might improve your post. Anyway, your formula appears to fail if $a<0$, it doesn't hold for e.g. $n=1$ and $(a,b)=(-1,1)$, for which the integral is $0$ but your formula yields $1$. – user170231 Oct 18 '23 at 16:13
  • thanks for the suggestion but the result holds for ${a} \lt{0}$ as the original question also has the same condition – Neelamber Mishra Oct 18 '23 at 17:08
  • and sorry for the bad framing of the question , as it was my first time i had asked a question on the website it became kind of messy . – Neelamber Mishra Oct 18 '23 at 17:24
  • What is ${x}$? Am I the only one reading this that does not know? – AlgTop1854 Oct 18 '23 at 18:38
  • @AlgTop1854 - it means the fractional part of $x$. I.e. $x$ minus the greatest integer less than $x$: $$x = \lfloor x \rfloor + {x}$$ – Paul Sinclair Oct 18 '23 at 19:02
  • Thanks - apparently there are multiple ways to define this for $x<0$ which matters for the above integral, so is there one most commonly used? – AlgTop1854 Oct 18 '23 at 22:06

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