The integral of $e^{x^2}$ is a well-known problem in mathematics, and unfortunately, it cannot be expressed in terms of elementary functions (polynomials, exponentials, logarithms, etc.). This means there is no simple closed-form solution.

However, the integral can be represented using a special function known as the error function, denoted as $\text{erf}(x)$, which is closely related to the Gaussian integral.

The integral is:

$\int e^{x^2} \, dx = \frac{\sqrt{\pi}}{2} \, \text{erfi}(x)$

where $\text{erfi}(x)$ is the imaginary error function, and it is related to the standard error function $\text{erf}(x)$.

Alternatively, for practical purposes, the integral is often computed numerically, using methods like series expansion or specialized computational algorithms.

Would you like a more detailed explanation on error functions, or how to evaluate this integral numerically?

Here are 5 related questions you might find useful:

1. What is the error function, and how is it derived?
2. How can I approximate the integral of $e^{x^2}$ numerically?
3. What are some other integrals that cannot be expressed in elementary functions?
4. How does the error function relate to the normal distribution in statistics?
5. Can the integral of $e^{x^2}$ be solved using a series expansion?

Tip: For integrals that don't have elementary solutions, numerical methods and special functions are often the go-to approach.