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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.9.112c
Chapter 8, Problem 8.9.112c

Gaussians An important function in statistics is the Gaussian (or normal distribution, or bell-shaped curve), f(x) = e^(-ax²).
c. Complete the square to evaluate ∫ from -∞ to ∞ of e^(-(ax² + bx + c)) dx, where a > 0, b, and c are real numbers.

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Start with the integral \( \int_{-\infty}^{\infty} e^{-(ax^2 + bx + c)} \, dx \), where \(a > 0\), and \(b, c\) are real numbers.
To simplify the exponent, complete the square for the quadratic expression \(ax^2 + bx + c\). Factor out \(a\) from the terms involving \(x\): \(ax^2 + bx + c = a\left(x^2 + \frac{b}{a}x\right) + c\).
Next, complete the square inside the parentheses: \(x^2 + \frac{b}{a}x = \left(x + \frac{b}{2a}\right)^2 - \left(\frac{b}{2a}\right)^2\). Substitute this back to get \(ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c\).
Rewrite the integral using the completed square form: \(\int_{-\infty}^{\infty} e^{-\left[a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b}{2a}\right)^2 + c\right]} \, dx = \int_{-\infty}^{\infty} e^{-a\left(x + \frac{b}{2a}\right)^2} e^{a\left(\frac{b}{2a}\right)^2 - c} \, dx\).
Since \(e^{a\left(\frac{b}{2a}\right)^2 - c}\) is a constant with respect to \(x\), factor it out of the integral. Then, perform the substitution \(u = x + \frac{b}{2a}\), which does not change the limits of integration because they are infinite. The integral reduces to \(e^{a\left(\frac{b}{2a}\right)^2 - c} \int_{-\infty}^{\infty} e^{-a u^2} \, du\), which is a standard Gaussian integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus a constant. This simplifies integration and other operations by transforming the expression into a form like a(x + d)² + e, making it easier to handle exponential functions involving quadratics.
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Gaussian Integral

The Gaussian integral refers to the integral of the function e^(-ax²) over the entire real line, which evaluates to √(π/a) for a > 0. This result is fundamental in probability and statistics, especially for normal distributions, and serves as a basis for evaluating more complex integrals involving quadratic exponents.
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Properties of Exponential Functions

Exponential functions with quadratic exponents, such as e^(-(ax² + bx + c)), can be manipulated using algebraic techniques like completing the square. Understanding how to factor and rewrite these functions is essential for integrating them, as it allows the integral to be expressed in terms of known Gaussian integrals.
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Related Practice
Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

48. ∫(0 to π/4) (1/(1 + x²)) dx; n = 64

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

The Eiffel Tower Property Let R be the region between the curves y = e^(-c·x) and y = -e^(-c·x) on the interval [a, ∞), where a ≥ 0 and c > 0.

The center of mass of R is located at (x̄, 0), where x̄ = [∫(a to ∞) x·e^(-c·x) dx] / [∫(a to ∞) e^(-c·x) dx]

(The profile of the Eiffel Tower is modeled by these two exponential curves; see the Guided Project "The exponential Eiffel Tower")

d. Prove this property holds for any a ≥ 0 and c > 0:

The tangent lines to y = ±e^(-c·x) at x = a always intersect at R's center of mass

(Source: P. Weidman and I. Pinelis, Comptes Rendu, Mechanique, 332, 571-584, 2004)

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Textbook Question

82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.

c. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, b]. Because this area depends on a and b, we call it A(a, b).

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Textbook Question

45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.

47. ∫(1 to e) (1/x) dx; n = 50

c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.

Textbook Question

77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly

Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).

d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.

Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

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