How do I evaluate Stieltjes-integral with sgn(\sin x)?

Question

How do I calculate the integral Stieltjes integral of:

$$\int_{-\pi}^{\pi} (x+2) d(x*sign(\sin x))$$

I know that $\int f(x) dg(x) = \int f(x)g(x)'dx$

But does the derivative $[sign(\sin x)]'=0$? But then how do I decide?

Answer 1

Answer to the question before it got edited: $sign(\sin \, x)=1$ for $x$ between $0$ and $\pi$ and $-1$ between $-\pi$ and $0$. The integral of $f$ w.r.t this function is simply $f(0)$. So the answer is $2$.

Answer to the question after it was edited: $x\, sign (sin\, x)=-x$ for $x \leq \pi$ and $x$ for $x \geq \pi$. There is no discontinuity at $0$. Hence the given integral is $-\int_{-\pi}^{0} (x+2)dx+\int_0^{\pi} (x+2)dx=\pi^{2}$.