Exercise: Integration by Parts for Stieltjes Integrals

Prerequisites: Riemann-Stieltjes Integral

Problem

Let $f(x)=x$ and $g(x)=x^2$ on $[0,1]$. Verify $\int_0^1 f\,dg+\int_0^1 g\,df=f(1)g(1)-f(0)g(0)$.

Hint