Evaluating $\int x f(x)\ \mathrm{d}f(x)$

Question

$$ \int xf(x)\ \mathrm{d}f(x) $$ I firstly thought we can assume $x$ as a constant but it isn't $f(x)$ depends on $x$ therefore $x$ depends on $f(x)$ . I just couldn't find a way out. If we say $f(x)=x^2$ what would be the result?

Answer 1

Use that

$$x^2 = \int_0^x 2y dy \Rightarrow df(x) = d(x^2) = 2xdx$$

If you don’t understand this, then you might read on lebesgue stieltjes integral.

Answer 2

If we say $f(x)=x^2$ we will have to evaluate the following integral: $$\int x\cdot x^2 \mathrm {d}(x^2)$$ This can be simplified as below: $$= \int x^3 \mathrm{d}(x^2)$$ $$= \int x^3\cdot 2x \mathrm{d}x$$ $$= 2\int x^4 \mathrm{d}x$$ $$= \frac25 \cdot x^5 + \mathrm {C}$$ Where $\mathrm{C}$ is the constant of integration

Answer 3

This is a Riemann-Stieltjes integral. If we assume that $f$ is continuously differentiable, we have that $$\int f(x) x \mathrm{d}f(x)=\int f(x) x f'(x) \mathrm{d}x$$