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What does $\int(x^2-4)d \lfloor x \rfloor$ mean? How do I integrate wrt $\lfloor x\rfloor$? This has to be a typo. enter image description here

ZSMJ
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    It means a Riemann-Stieltjes integral. – Jakobian Jun 28 '19 at 17:25
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    Possible duplicate of Integration of a function with respect to another function. – SlipEternal Jun 28 '19 at 17:33
  • @Jakobian Could you link me to a definition of RS Integrals that uses that notation? The wiki article doesn't seem to. – ZSMJ Jun 28 '19 at 17:36
  • Where are these questions from? – Ovi Jun 28 '19 at 17:37
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    @rhaldryn it's on the wiki. $f(x) = x^2-4$, $g(x) = \lfloor x \rfloor$, $\int_0^4 f(x) dg(x)$ – Jakobian Jun 28 '19 at 17:39
  • @Ovi It's an M.Sc. math entrance exam of a university in India. However Riemann-Stieltjes integrals were not in the syllabus. – ZSMJ Jun 28 '19 at 17:40
  • @rhaldryn Ah thanks. For what it's worth, the only time I have seen the notation $\int f(x) d$(something other than $x$) is in Riemann-Stieltjes integral, so I'm not sure, maybe the syllabus was mistaken. I was actually asking about the problems because they look interesting and would be good practice for me. Would you happen to have a link to the problems, or know where I could get more such problems? Thanks. – Ovi Jun 28 '19 at 17:44
  • You can go to any state university or institute (in India at least eg, IIT, IMSC, CMI, ISI, TIFR etc.) website and search for the previous year question papers. Most of them take MCQ tests like these. If you want proof based subjective questions then there are tons of problem books recommendations on MSE. – ZSMJ Jun 28 '19 at 17:53

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Working with Riemann-Stieltjes integrals,$$\int_0^4(x^2-4)d\lfloor x\rfloor=\left[(x^2-4)\lfloor x\rfloor\right]_0^4-\int_0^4\lfloor x\rfloor 2xdx\\=48-\int_1^22xdx-2\int_2^32xdx-3\int_3^42xdx\\=48-(2^2-1^2)-2(3^2-2^2)-3(4^2-3^2)=48-3-10-21=14.$$ More generally,$$\int_{0}^{n}f\left(x\right)d\left\lfloor x\right\rfloor =nf\left(n\right)-\sum_{k=1}^{n-1}k\left(f\left(k+1\right)-f\left(k\right)\right)=\sum_{k=1}^{n}f\left(k\right)$$for $n\in\mathbb{N}$.

J.G.
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You can use integration by parts as follows. $$\int_0^4 (x^2-4)d\lfloor x\rfloor = (x^2-4)\lfloor x\rfloor |_{x=0}^{x=4} - \int_0^4 \lfloor x\rfloor d(x^2-4) = \\ 48-\int_0^4 2x\lfloor x\rfloor dx = 48-(4-1)-2(9-4)-3(16-9) = 14 $$

Jakobian
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