What does $\int(x^2-4)d \lfloor x \rfloor$ mean?

Question

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.

Answer 1

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 $n\in\mathbb{N}$.

Answer 2

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