Textbook Question
88. The region in Exercise 87 is revolved about the x-axis to generate a solid.
a. Find the volume of the solid.
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.1.54a
Using different substitutions
Show that the integral
∫((x² - 1)(x + 1))^(-2/3) dx
can be evaluated with any of the following substitutions.
a. u = 1/(x + 1)
What is the value of the integral?
Verified step by step guidance1
Start with the integral \( \int ((x^2 - 1)(x + 1))^{-2/3} \, dx \). First, simplify the expression inside the integral. Notice that \( x^2 - 1 = (x - 1)(x + 1) \), so the integrand becomes \( ((x - 1)(x + 1)^2)^{-2/3} \).
Rewrite the integrand as \( (x - 1)^{-2/3} (x + 1)^{-4/3} \) by distributing the exponent \( -2/3 \) to each factor inside the parentheses.
Use the substitution \( u = \frac{1}{x + 1} \). Then, express \( x + 1 \) and \( dx \) in terms of \( u \): \( x + 1 = \frac{1}{u} \) and \( dx = -\frac{1}{u^2} du \) (found by differentiating \( u = \frac{1}{x + 1} \) with respect to \( x \)).
Rewrite \( x - 1 \) in terms of \( u \) using \( x = \frac{1}{u} - 1 \), so \( x - 1 = \frac{1}{u} - 2 \). Substitute these expressions into the integrand and the differential \( dx \) to rewrite the entire integral in terms of \( u \).
Simplify the resulting integral in \( u \) and then integrate using appropriate techniques (such as power rule or further substitution). After integration, substitute back \( u = \frac{1}{x + 1} \) to express the answer 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.
Substitution Method in Integration
The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. By letting u equal a function of x, we rewrite the integral in terms of u and du, making it easier to evaluate. This technique is especially useful when the integral contains composite functions.
Recommended video:
07:33
Euler's Method
Algebraic Manipulation of Integrands
Before applying substitution, it is important to simplify or rewrite the integrand using algebraic identities or factorization. This helps identify suitable substitutions and reduces the complexity of the integral. For example, expressing (x² - 1)(x + 1) in a simpler form can clarify the substitution process.
Recommended video:
05:22Completing the Square to Rewrite the Integrand
Integration of Power Functions
Integrals involving expressions raised to fractional powers, such as (-2/3), require understanding how to integrate power functions. After substitution, the integral often reduces to a form ∫u^n du, which can be integrated using the power rule: ∫u^n du = u^(n+1)/(n+1) + C, provided n ≠ -1.
Recommended video:
07:32Representing Functions as Power Series
Related Practice
Textbook Question
Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.
a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?
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Textbook Question
81. Find the values of p for which each integral converges.
a. ∫ from 1 to 2 of [dx / (x (ln x)^p)]
Textbook Question
Consider the region bounded by the graphs of
y = ln(x), y = 0, and x = e.
a. Find the area of the region.
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Textbook Question
Finding area
Find the area of the region enclosed by the curve y = x cos(x) and the x-axis (see the accompanying figure) for:
a. π/2 ≤ x ≤ 3π/2.
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Textbook Question
In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (a) the Trapezoidal Rule (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from -1 to 1 of (x² + 1) dx