# calculus and analysis – How does mathematica integrate around points where the integrand evaluates to ”indeterminate?”

I am interested in studying a numerical integral where the integrand, when evaluated naively at some points in the domain of integration, is indeterminate. However, when one carefully takes limits to these points, the integrand turns out to be finite. Since my actual integrand is multi-dimensional and complicated, let me pose my question in the context of a simple example instead. If I ask mathematica to evaluate $$frac{sin x}{x}$$ at $$x=0$$, it says it’s indeterminate (although we know that the actual value as $$x to 0$$ is finite). However, if I evaluate the numerical integral of this function from $$-1 leq x leq 1$$, it gives me the right result, i.e, a value that matches the result of analytic integration.

So my question is: in numerical integration, how does mathematica deal with points where if you ask mathematica to evaluate the integrand, it gives ”indeterminate.” Are these points omitted? If not, how are they accounted for? In the example above, mathematica was somehow able to deal with $$x=0$$ correctly when integrating numerically even though it said that the function evaluated at $$x=0$$ was indeterminate.