Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.5.82

Capitolo 8, Problema 8.5.82

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.
82. ∫ [dx / (x√(1 + 2x))]

Guida verificata passo dopo passo

Identify the integral to solve: \(\int \frac{dx}{x \sqrt{1 + 2x}}\).

Recognize that the expression under the square root, \(1 + 2x\), suggests a substitution to simplify the radical. Let \(t = \sqrt{1 + 2x}\), so that \(t^2 = 1 + 2x\).

Differentiate both sides of \(t^2 = 1 + 2x\) with respect to \(x\) to find \(dx\) in terms of \(dt\): \(2t \frac{dt}{dx} = 2\), which implies \(\frac{dt}{dx} = \frac{1}{t}\), so \(dx = t \, dt\).

Express \(x\) in terms of \(t\) from \(t^2 = 1 + 2x\): \(x = \frac{t^2 - 1}{2}\). Substitute \(x\) and \(dx\) back into the integral to rewrite it entirely in terms of \(t\).

Simplify the integral after substitution and then proceed to use partial fractions or other integration techniques as appropriate to evaluate the integral.

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Substitution Method (Change of Variables)

This technique involves replacing a complicated expression with a simpler variable to make integration easier. For example, setting u = 1 + 2x transforms the integral into a function of u, simplifying the square root and rational expressions. It is often the first step before applying other integration methods.

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Substitution With an Extra Variable

Partial Fraction Decomposition

Partial fractions break down a complex rational function into simpler fractions that are easier to integrate. After substitution, the integrand often becomes a rational function suitable for this method, allowing the integral to be expressed as a sum of simpler terms.

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Percorso guidato

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Partial Fraction Decomposition: Distinct Linear Factors

Integration of Rational Functions Involving Roots

Integrals containing roots like √(1 + 2x) often require manipulation to rewrite the integrand in a rational form. Understanding how to handle expressions with roots and convert them into integrable forms is essential, often combining substitution and partial fractions.

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Integrals Involving Natural Logs: Substitution

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7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

31. ∫ √(x² - 8x) dx, x > 8

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63. (Use of Tech) Normal distribution of heights

The heights of U.S. men are normally distributed with a mean of 69 in and a standard deviation of 3 in. This means that the fraction of men with a height between a and b (with a < b) inches is given by the integral

(1/(3√(2π))) ∫ₐᵇ e^(-((x-69)/3)²/2) dx.

What percentage of American men are between 66 and 72 inches tall? Use the method of your choice, and experiment with the number of subintervals until you obtain successive approximations that differ by less than 10⁻³.

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9–61. Trigonometric integrals Evaluate the following integrals.

16. ∫ sin²θ cos⁵θ dθ

132

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29-34. {Use of Tech} Comparing the Midpoint and Trapezoid Rules

Apply the Midpoint and Trapezoid Rules to the following integrals. Make a table similar to Table 8.5 showing the approximations and errors for n = 4, 8, 16, and 32. The exact values of the integrals are given for computing the error.

33. ∫(0 to π) sin x cos(3x) dx = 0

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7–84. Evaluate the following integrals.

33. ∫ [eˣ / (a² + e²ˣ)] dx, where a ≠ 0

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7–64. Integration review Evaluate the following integrals.

12. ∫ from -5 to 0 of dx / √(4 - x)

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