There is a well-known theorem that states that for $f$ continuous and $f,\hat f$ integrable,
$$f(0)=\lim_{T\to\infty}\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx.$$
So it should be that
$$\int_{-\infty}^\infty f(x)\frac{\sin(Tx)}{x}dx=f(0)+\text{(Error term)}.$$
Are there known explicit formulas for estimating this error term?