An antiderivative $F(x)$ of the function $f(x) = e^{-x^2}$ is given by:

$F(x) = \int_{-3}^{x} e^{-t^2} \, dt + 1$

I determined this formula by using the fact that $F(-3) = 1$ to define the antiderivative as a definite integral with the lower limit $-3$, ensuring the condition $F(-3) = 1$ is satisfied by adding 1 to the integral.

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How can we approximate the integral $\int e^{-x^2} dx$ using numerical methods?
2. What is the error function (erf), and how is it related to integrals of $e^{-x^2}$?
3. What is the significance of Gaussian integrals in probability theory?
4. How can we express an indefinite integral of $e^{-x^2}$ in terms of special functions?
5. How does the antiderivative of $e^{-x^2}$ behave as $x \to \infty$?

Tip: The function $e^{-x^2}$ does not have an elementary antiderivative, so integrals involving it often involve special functions or numerical approaches.