Textbook Question
4. What substitutions are made to evaluate integrals of sin(mx)sin(nx), sin(mx)cos(nx), and cos(mx)cos(nx)? Give an example of each case.
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- 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
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 2 of 1/s² ds
Textbook Question
Evaluate the integrals in Exercises 33–36.
∫ [1 / (x(9 - x²))] dx
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
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
Textbook Question
Evaluate ∫ sec θ dθ by:
a. Multiplying by (sec θ + tan θ) / (sec θ + tan θ) and then using a u-substitution.