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.
 A: This is just the Riemann-Stieltjes integral. If you have monotone function $g$ which is continuously differentiable (and the range of integration is a compact interval $[a,b]$), then $$\int_a^b f(x) dg(x) = \int^b_a f(x)g'(x) dx.$$ Here we have $$\int^2_1 x d[x^2] = \int^2_1 2x^2 dx = \frac 2 3(2^3 - 1^3) = \frac {14} 3$$
For more info: https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral
EDIT: I see now that we are using $g(x) = [x^2]$ as the integrator where $[\cdot]$ is the floor function. Generally, if $f$ is continuous on $[a,b]$ and $g$ is piecewise constant with jump discontinuities at $a_1, a_2, \ldots, a_\ell$ in the interval (but $g$ has one-sided limits $g(a_j^+)$ and $g(a_j^-)$ on each side of the jumps), then a similar argument will give $$\int^b_a f(x) dg(x) = \sum_{j=1}^\ell f(a_j) (g(a_j^+) - g(a_j^-)).$$ This gives $$\int^2_1 x d[x^2] = \sqrt 2 + \sqrt 3 + \sqrt 4.$$
A: One has
$$
\int_1^2 x\,d[x^2]=\sqrt{2}+\sqrt{3}+\sqrt{4}.
$$
Argument 1
We use

Theorem (9-11 in Apostol's Real Analysis) Let $\alpha$ be a step function defined on $[a,b]$ with jump $\alpha_k$ at $x_k$, where $a\leq x_1<x_2<\ldots<x_n\leq b$. Let $f$ be defined on $[a,b]$ in such a way that not both $f$ and $\alpha$ are discontinuous from the right or from the left at each $x_k$. Then $\int_a^b f\,d\alpha$ exists and
  $$
\int_a^b f(x)\,d\alpha(x)=\sum_{k=1}^n f(x_k)\alpha_k.
$$

If we take $f(x)=x$ and $\alpha(x)=[x^2]$ the conditions are fulfilled, and with $x_1=\sqrt{2}$, $x_2=\sqrt{3}$ and $x_3=\sqrt{4}$, we find that
$$
\int_1^2x\,d[x^2]=\sqrt{2}(2-1)+\sqrt{3}(3-2)+\sqrt{4}(4-2)=\sqrt{2}+\sqrt{3}+\sqrt{4}.
$$
Argument 2
We can use integration by parts. The general formula reads
$$
\int_a^b f(x)\,d\alpha(x)+\int_a^b \alpha(x)\,df(x)=f(b)\alpha(b)-f(a)\alpha(a).
$$
With $f(x)=x$ and $\alpha(x)=[x^2]$, we have
$$
\begin{aligned}
\int_1^2 x\,d[x^2]&=-\int_1^2 [x^2]\,dx+2[2^2]-1[1^2]\\
&=-\bigl\{(\sqrt{2}-1)+2(\sqrt{3}-\sqrt{2})+3(\sqrt{4}-\sqrt{3})\bigr\}+8-1\\
&=\sqrt{2}+\sqrt{3}+\sqrt{4}.
\end{aligned}
$$
A: 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).
