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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 8.7.31
Chapter 8, Problem 8.7.31

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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1
Rewrite the expression inside the square root to make it easier to work with. Complete the square for the quadratic expression x² - 8x. Completing the square involves rewriting x² - 8x as (x - 4)² - 16.
Substitute the completed square form into the integral: ∫ √((x - 4)² - 16) dx.
Recognize that this integral resembles a standard form found in a table of integrals, specifically one involving √(u² - a²). To match the table form, let u = x - 4 and du = dx.
Rewrite the integral in terms of u: ∫ √(u² - 16) du. Now the integral is in a standard form that can be evaluated using a table of integrals.
Use the appropriate formula from the table of integrals for ∫ √(u² - a²) du, where a² = 16. Substitute back u = x - 4 into the result to express the solution in terms of x.

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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 transform a quadratic expression into a perfect square trinomial. This method is essential for simplifying integrals involving quadratic terms, as it allows for easier integration by rewriting the expression in a more manageable form. For example, the expression x² - 8x can be rewritten as (x - 4)² - 16, facilitating the integration process.
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Completing the Square to Rewrite the Integrand

Indefinite Integrals

Indefinite integrals represent a family of functions whose derivative gives the integrand. They are expressed without limits and include a constant of integration, typically denoted as 'C'. Understanding how to evaluate indefinite integrals using techniques such as substitution or reference to integral tables is crucial for solving problems in calculus, including those that require preliminary transformations.
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Integral Tables

Integral tables are collections of standard integrals that provide quick references for evaluating common integrals without performing the integration from first principles. They are particularly useful for complex functions or those that require specific techniques, such as trigonometric or exponential substitutions. Familiarity with these tables can significantly expedite the process of solving integrals, especially when combined with preliminary algebraic manipulations.
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Related Practice
Textbook Question

7–84. Evaluate the following integrals.

16. ∫ [1 / (x⁴ – 1)] dx

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Textbook Question

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

98. ∫(from 0 to 1) (ln x)/(1+x) dx = -π²/12

Textbook Question

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.

Textbook Question

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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Textbook Question

7–84. Evaluate the following integrals.

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

Textbook Question

87-92. An integrand with trigonometric functions in the numerator and denominator can often be converted to a rational function using the substitution u = tan(x/2) or, equivalently, x = 2 tan⁻¹u. The following relations are used in making this change of variables.

A: dx = 2/(1 + u²) du

B: sin x = 2u/(1 + u²)

C: cos x = (1 - u²)/(1 + u²)

91. Evaluate ∫[0 to π/2] dθ/(cos θ + sin θ).