I know that the domain of the function $F(x)=\int_0^x(te^{-\frac{1}{t^2}})dt$ is the set of $x$ such as $F(x)$ is integrable.
The function $f(x)=te^{-\frac{1}{t^2}}$ is continuous in $\mathbb{R}$, so first of all I have to prove that $F(x)$ is integrable as $x\rightarrow 0^+$, which is true.
How do I calculate the complete domain of $F(x)$?
Update: maybe I should solve the integral as $\lim_{m\to \infty}\int_0^mf(x)$ and see for which $m$ the integral converges.
The integral is finite for every $x$ so the domain is $\mathbb R$. [For $x<0$ $F(x)=-\int_x^{0} te^{-1/t^{2}} \, dt$].