The function e^(x^2) is not integrable in terms of elementary functions. This means that you cannot express the integral of e^(x^2) using a finite combination of basic functions like polynomials, trigonometric functions, exponentials, and logarithms.
To understand why, we can look at the nature of the function itself. The exponential function grows very quickly, and when it’s raised to the power of x^2, it grows even faster as x moves away from zero. When attempting to compute the integral of e^(x^2), say from negative infinity to positive infinity, we find that the area under the curve does not yield a simple form that can be expressed using basic mathematical functions.
Furthermore, this integral is closely related to the error function, which is a special function used extensively in probability and statistics, particularly in normal distributions. The error function, denoted as erf(x), is not elementary itself but helps evaluate integrals like this one, specifically the integral of e^(-x^2), which does converge and is integrable.
In summary, e^(x^2) is non-integrable in the elementary sense because it cannot be expressed in a simple closed form and leads to the necessity of special functions for its evaluation.