1. **Problem Statement:** Evaluate the integral $$\int e^{x^2} \, dx$$ which is known to be a difficult integral with no elementary antiderivative. 2. **Explanation:** The function $$e^{x^2}$$ does not have an antiderivative expressible in terms of elementary functions (polynomials, exponentials, logarithms, trigonometric functions, etc.). 3. **Approach:** We use the special function called the error function, defined as $$\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} dt$$. 4. **Transform the integral:** To relate $$\int e^{x^2} dx$$ to known functions, consider the substitution $$x = i t$$ (where $$i$$ is the imaginary unit), then $$e^{x^2} = e^{-t^2}$$. 5. **Result:** The integral can be expressed in terms of the imaginary error function $$\operatorname{erfi}(x)$$ defined as $$\operatorname{erfi}(x) = -i \operatorname{erf}(i x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{t^2} dt$$. 6. **Final answer:** $$\int e^{x^2} dx = \frac{\sqrt{\pi}}{2} \operatorname{erfi}(x) + C$$ where $$C$$ is the constant of integration. This shows the integral cannot be expressed in elementary terms but can be represented using special functions.