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$[x]$ denotes the greatest integer $\leq x$. Let $f(x)=[x]$

What is the indefinite integral (without limits) of $f(x)$ ?

And if $f(x) = e^{-7[x]}$ what would the indefinite integral be?

Gin99
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    $[x]$ can be thought of as a sum of step functions, so the integral will likely be a sum of ever-increasing ramp functions. – AspiringMathematician Apr 25 '17 at 05:09
  • As-is, this function cannot have an antiderivative as a consequence of Darboux's theorem, because $[x]$ does not satisfy the intermediate value property (see below). What you want can only be done if we restrict $f$ to the points $\mathbb{R} \setminus \mathbb{Z}$, and in this case the antiderivative would be defined piecewise. As Guilherme says, it will look like a union of lines with different slopes. Think about what the derivative of $\text{abs}(x)$ will look like for $x \neq 0$ to get some intuition. https://en.wikipedia.org/wiki/Darboux%27s_theorem_(analysis) – Kaj Hansen Apr 25 '17 at 05:10
  • Lots of possibly relevant questions and answers if you search this site for the title of your question. – Ethan Bolker Apr 25 '17 at 14:41
  • @KajHansen then how did one came to a result like the one below? – Gin99 Apr 25 '17 at 22:20
  • See https://math.stackexchange.com/questions/1312213/indefinite-integral-of-floor-function-integration-by-substitution – John Wayland Bales Apr 26 '17 at 05:10
  • I found this [link] https://math.stackexchange.com/questions/33547/is-this-a-justified-expression-for-the-integral-of-the-floor-function/33568#33568. In the answer of the thread in the link, does anyone one knows how did he get the $\{x\}\lfloor x\rfloor$ part? Or should I open another thread just to ask this? – Gin99 Apr 26 '17 at 06:20

1 Answers1

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$\int[x]~dx=x[x]-\dfrac{[x]([x]+1)}{2}+C$

Harry Peter
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  • Why is the integral different than the one posted here https://math.stackexchange.com/questions/33547/is-this-a-justified-expression-for-the-integral-of-the-floor-function/33568#33568 – Gin99 Apr 26 '17 at 06:25