Question

Identify each of the following integrals or expressions as one of the functions of this chapter. Check your work by evaluating both your answer and the original problem by computer. Be sure you understand your computer program's notation. $$\int_{1}^{\infty} e^{-x^{2} / 2} d x$$

Short answer

The integral \(\int_{1}^{\infty} e^{-x^{2} / 2} d x\) transforms to an equivalent \ \sqrt{2} \int_{1/\sqrt{2}}^{\infty} e^{-u^2} du \, which can be computed using the error function complement \text{erfc}(x)\.

Step-by-step solution

Recognize the Integral Form

The given integral is \(\int_{1}^{\infty} e^{-x^{2} / 2} d x\). This resembles the Gaussian integral but with different limits and a variable change.

Standard Gaussian Integral

Recall that the standard Gaussian integral \(\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}\). Our integral is similar but involves \ e^{-x^{2} / 2} \ and different limits.

Substitution for Simplification

Make a substitution to match the standard form. Let \ u = \frac{x}{\sqrt{2}} \ so that \ du = \frac{1}{\sqrt{2}} dx \ and thus \ dx = \sqrt{2} du\. The integral becomes \ \(\int_{1}^{\infty} e^{-x^{2} / 2} d x = \sqrt{2} \int_{1/\sqrt{2}}^{\infty} e^{-u^2} du\).\

Evaluate the Incomplete Gamma Function

This integral is an instance of the incomplete gamma function represented as \ \(\text{erfc}(x)\). The complement of the error function \ \text{erf}(x) \ calculates the specific integral result.

Utilize Computational Tools

Using a computer algebra system (CAS) like Mathematica or a scientific calculator, the incomlete gamma function can be evaluated. Enter the integral or equivalent erfc function into the software.

Key concepts

Gaussian function

The Gaussian function is a fundamental function in mathematics and physics, often used to describe bell-curve shaped data or phenomena. It is defined as \[ f(x) = a e^{-\frac{(x - b)^2}{2c^2}} \]where:

• \(a\) is the peak amplitude,
• \(b\) is the position of the center of the peak,
• \(c\) controls the width of the curve.

One of the most common uses of the Gaussian function is in the Gaussian integral: \[ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}\]which is central to probability theory and statistics. In the given problem, the integral \( \int_{1}^{\infty} e^{-x^{2} / 2} dx \)resembles a Gaussian function but needs adjustment for proper evaluation.

substitution method

The substitution method is a technique used to simplify integrals by changing variables. It helps transform a complex integral into a more manageable form. For example, in the given integral \(\int_{1}^{\infty} e^{-x^{2} / 2} dx \), we use the substitution: \( u = \frac{x}{\sqrt{2}} \). This implies that \(dx = \sqrt{2} du \), shifting the integral limits from \(x\to u\). Therefore, the integral transforms into: \[ \int_{1/\sqrt{2}}^{\infty} e^{-u^2} du \] This form looks more like our standard Gaussian integral, and simplifies evaluation using known functions or computational tools.

incomplete gamma function

The incomplete gamma function is a generalization of the gamma function, and is useful for evaluating integrals with varying upper or lower limits. The given integral \( \int_{1}^{\infty} e^{-x^{2} / 2} dx \)cannot be solved with standard elementary functions. However, it connects to the incomplete gamma function. Specifically, the integral \( \int_{1/\sqrt{2}}^{\infty} e^{-u^2} du \)is a part of the incomplete gamma function. Recognizing and utilizing this relationship allows us to use computational tools for evaluation.

error function (erf)

The error function (erf) is another important function related to the Gaussian function. It is defined as: \[ \text{erf}(x) = \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^2} dt\]The error function helps with the evaluation of integrals of Gaussian functions over various intervals. The complement of this function, called erfc(x), is particularly useful in our given problem: \( \int_{1}^{\infty} e^{-x^{2} / 2} dx \). This integral correlates to \(\frac{\text{erfc}(u)}{2}\) after substitution. Utilizing computational tools, we solve for the desired value using erfc(x). Thus, simplifying complex integrals becomes possible.