TITLE: Double integral of $\exp(-t^2)$ on a triangular domain
QUESTION [5 upvotes]: My calculus teacher gave us this interesting problem: Calculate
$$ \int_{0}^{1}F(x)\,dx,\ $$ where $$F(x) = \int_{1}^{x}e^{-t^2}\,dt $$
The only thing I can think of is using the Taylor series for $e^{-t^2}$ and go from there, but since we've never talked about uniform convergence and term by term integration, I suppose that there is an easier way to do this.

REPLY [7 votes]: No fancy stuff is needed. I think you could just integrate by parts.
$$\int_0^1 F(x)\,dx=[xF(x)]_0^1-\int_0^1 xF'(x)\,dx$$
The outintegrated part cancel, and using the fundamental theorem of calculus, $F'(x)=e^{-x^2}$. Thus
$$\int_0^1 F(x)\,dx=-\int_0^1 xe^{-x^2}\,dx
$$
from where I think you can finish.