Manipulating integrals containing the floor function

Question

Since $x-\lfloor x\rfloor$ is periodic, for $x\notin\mathbb{Z}$, we can derive the following Fourier series $$\lfloor x\rfloor=x-\frac{1}{2}+\frac{1}{\pi}\sum_{k=1}^\infty\frac{\sin(2\pi kx)}{k},\quad x\notin\mathbb{Z}.$$ Is it then legal to substitute this into integrals in the following way? $$\int dx\rho(x)\lfloor x\rfloor=? \int dx\rho(x)\left(x-\frac{1}{2}+\frac{1}{\pi}\sum_{k=1}^\infty\frac{\sin(2\pi kx)}{k}\right), \quad \rho \quad\text{generic.}$$ In the application that I'm considering, I would then want to swap the order of summation and integration by Fubini's theorem.