1. The problem is to find the integral of the function $e^{x^2}$. 2. The integral we want to solve is $$\int e^{x^2} \, dx.$$ 3. Unfortunately, there is no elementary antiderivative for $e^{x^2}$, meaning it cannot be expressed in terms of basic functions like polynomials, exponentials, logarithms, or trigonometric functions. 4. To handle this integral, mathematicians use a special function called the error function, denoted as $\operatorname{erf}(x)$, or define it as a non-elementary integral. 5. The integral $$\int e^{x^2} \, dx$$ is typically expressed in terms of the imaginary error function or evaluated numerically. 6. Therefore, the integral of $e^{x^2}$ does not have a simple closed form and is left as $$\int e^{x^2} \, dx.$$