Calculating Integral E X 2

The integral of e^x^2 is a fundamental calculation in calculus that appears in probability, physics, and engineering. This guide explains how to compute it, provides a working calculator, and includes practical examples.

What is the integral of e^x^2?

The integral of e^x^2 is expressed as:

∫ e^(x²) dx = (√π/2) * erf(x) + C

Where:

erf(x) is the error function, defined as erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt
C is the constant of integration

The error function is a special function that cannot be expressed in terms of elementary functions, so the integral of e^x^2 is typically left in terms of erf(x).

How to calculate the integral of e^x^2

Step-by-step method

Recognize that the integral of e^(x²) is not elementary and requires special functions.
Use integration by parts or substitution to express the integral in terms of the error function.
Recall the standard result: ∫ e^(x²) dx = (√π/2) * erf(x) + C
Apply the limits of integration if definite integral is required.

Assumptions and limitations

The integral of e^x^2 is an improper integral that converges to infinity at both ±∞. The error function erf(x) is defined for all real x and has the following properties:

erf(0) = 0
erf(∞) = 1
erf(-x) = -erf(x)

Note: The integral of e^x^2 cannot be expressed in terms of elementary functions. The error function is the standard way to represent this integral.

Examples of calculating the integral of e^x^2

Example 1: Indefinite integral

Calculate ∫ e^(x²) dx

Solution:

∫ e^(x²) dx = (√π/2) * erf(x) + C

Example 2: Definite integral from 0 to 1

Calculate ∫₀¹ e^(x²) dx

Solution:

∫₀¹ e^(x²) dx = (√π/2) * [erf(1) - erf(0)] = (√π/2) * erf(1)

Numerical approximation: erf(1) ≈ 0.8427, so the result is approximately 0.8862.

FAQ

Is the integral of e^x^2 elementary?

No, the integral of e^x^2 cannot be expressed in terms of elementary functions. It requires the error function erf(x).

What is the error function erf(x)?

The error function erf(x) is defined as erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt. It is a special function that appears in probability, statistics, and physics.

How do I compute the integral of e^x^2 numerically?

For numerical computation, you can use the error function approximation or numerical integration methods. The calculator on this page provides a convenient way to compute the integral.