Calculate a Riemann Stieltjes Integral $\int_0^3 x d([x] - x) = 3/2$.

Question

I need to prove that

$$\int_0^3 x d([x] - x) = \dfrac32$$

However I can not think of a change of variable that can be used so I tried to approach it using Riemann Sums $\sum_{i=0}^3(\alpha(i)-\alpha(i-1) )i$ But I can not prove that the result is 3/2. I took it from the Apóstol on Mathematical Analysis.

Answer 1

We discussed this type of integrals here, see the details there and you find it easy $$\int_0^3 x d([x] - x) =\int_0^3 x d[x] - \int_0^3xdx = 1+2+3-\dfrac92 = \dfrac32$$

Answer 2

By Wiki: Integration by parts for R-S, we have

$$\int_0^3 x d([x] - x) = 3 ([3] - 3) - 0 ([0] - 0) - \int_0^3 [x] - x dx$$

$$= 0 + 0 - \int_0^3 [x] - x dx$$

$$= \int_0^3 x - [x] dx,$$

an ordinary Riemann integral.