Evalute the Riemann-Stieltjes integral. (Here $x\mapsto [x]$ denotes the greatest integer functión).
$$\int_0^4x^2d([x^2])$$
My approach:
By integration by parts we have that: $$\int_0^4x^2d([x^2])=(4^2)([4^2])-\int_0^4[x^2]d(x^2) = (16)(15)-\int_0^4[x^2]d(x^2)$$ because $x^2$ is differtiable at $[0,4]$ can rewrite it like a Riemann integral: $$\int_0^4[x^2]d(x^2)= \int_0^4[t^2]2tdt$$ I´m kind of confused on how to evaluate $\int_0^4[t^2]2tdt$. Any suggestions would be great!