Textbook Question
Average value
In a mass-spring-dashpot system like the one in Exercise 65, the mass's position at time t is
y = 4e^(-t)(sin(t) - cos(t)), t ≥ 0.
Find the average value of y over the interval 0 ≤ t ≤ 2π.
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 8.4.61
Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.
Verified step by step guidance1
Identify the integral to solve: \(\int x^{3} \sqrt{1 - x^{2}} \, dx\) and recognize that the expression under the square root, \(1 - x^{2}\), suggests a trigonometric substitution involving sine or cosine.
Use the substitution \(x = \sin(\theta)\), which implies \(dx = \cos(\theta) \, d\theta\). This substitution transforms the square root as \(\sqrt{1 - x^{2}} = \sqrt{1 - \sin^{2}(\theta)} = \cos(\theta)\).
Rewrite the integral in terms of \(\theta\): replace \(x^{3}\) with \(\sin^{3}(\theta)\), \(\sqrt{1 - x^{2}}\) with \(\cos(\theta)\), and \(dx\) with \(\cos(\theta) \, d\theta\). The integral becomes \(\int \sin^{3}(\theta) \cos(\theta) \cdot \cos(\theta) \, d\theta = \int \sin^{3}(\theta) \cos^{2}(\theta) \, d\theta\).
Simplify the integral to \(\int \sin^{3}(\theta) \cos^{2}(\theta) \, d\theta\). Consider using a substitution such as \(u = \cos(\theta)\) or expressing powers of sine in terms of cosine to facilitate integration.
After integrating with respect to \(\theta\), substitute back \(\theta = \arcsin(x)\) 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.
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(a² - x²), √(a² + x²), or √(x² - a²). By substituting x with a trigonometric function (e.g., x = a sin θ), the radical expression transforms into a trigonometric identity, making the integral easier to evaluate.
Recommended video:
6:04
Introduction to Trigonometric Functions
Integral of Powers of x
Understanding how to integrate powers of x, such as x³, is essential. When combined with substitution, recognizing how to express x and dx in terms of θ allows the integral to be rewritten entirely in trigonometric terms, facilitating the integration process.
Recommended video:
05:58Intro to Power Series
Trigonometric Identities
Familiarity with fundamental trigonometric identities, like sin²θ + cos²θ = 1, is crucial. These identities help simplify the integrand after substitution, enabling the integral to be expressed in a solvable form involving basic trigonometric functions.
Recommended video:
7:17Verifying Trig Equations as Identities
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt
Textbook Question
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 2 sin^2(t) sec^4(t) dt
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ e^(z + eᶻ) dz
Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ tan^(-1)(x) / x² dx