Solve integral $\int_1^2x\,d[x^2]$ with floor differential. 
How can I evaluate the integral
  $$\int_1^2x\,d[x^2]$$

I think this meaningless, because the area below separate points is not defined.
Note: $[.]$ is the floor function.

Compute this as Riemann-Stieltjes integral: $$\int\limits_a^b f(x)\text{d}g(x)=\int\limits_a^b f(x)g'(x)\text{d}x$$ under suitable regularity asssumptions (which are in this case satisfied).
