What is the domain of the function $F(x)=\int_{0}^{x}\frac{\operatorname{arctan}(t)}{t}dt$? What is the domain of the following function?
$$F(x)=\int_{0}^{x}\frac{\operatorname{arctan}(t)}{t}dt$$
On the one hand, the internal function is not defined at 0, but on the other hand, it's defined on every other point except zero, so the region can be calculated.

The function can be extended in $x=0$ to obtain a continuous function defined on the whole real line. Hence such extended function is Riemann integrable and hence its integral on $[0,x]$ is well defined for all $x$. Now remember that if you modify a function in a finite number of points, its integral does not change. Hence you can extended your function with any value in $x=0$ and obtain an Riemann integrable function. 
However, strictly speaking, Riemann integrals on the interval $[a,b]$ is defined only for functions which are defined on the whole interval $[a,b]$. If the function is only defined on the interval $(a,b]$ this should be considered an improper integral i.e.:
$$
F(x) = \int_0^x f(t)\, dt = \begin{cases}\lim_{a\to 0^+}  \int_a^x f(t)\, dt & \text{if } x>0 \\ \lim_{a\to 0^-}\int_a^x f(t)\, dt & \text {if $x<0$ }\\ 0 &\text{if } x=0\end{cases}.
$$
But, again, since the function $f$ can be extended with continuity in $x=0$ the limit exists and is equal to the integral on $[0,x]$ of the extended function.
