Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos²(θ)) dθ
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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 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 8.4.14
Evaluate the integrals in Exercises 1–14.
∫ (2 dx) / (x³ √(x² - 1)), where x > 1
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{2 \, dx}{x^{3} \sqrt{x^{2} - 1}}\) with the condition \(x > 1\).
Recognize that the integrand contains \(\sqrt{x^{2} - 1}\), which suggests using a trigonometric substitution such as \(x = \sec(\theta)\) because \(\sec^{2}(\theta) - 1 = \tan^{2}(\theta)\).
Perform the substitution: let \(x = \sec(\theta)\), then compute \(dx = \sec(\theta) \tan(\theta) \, d\theta\). Also, rewrite the expressions inside the integral in terms of \(\theta\).
Rewrite the integral entirely in terms of \(\theta\) by substituting \(x\), \(dx\), and \(\sqrt{x^{2} - 1}\), then simplify the resulting expression to a form that is easier to integrate.
Integrate with respect to \(\theta\), then back-substitute using \(\theta = \sec^{-1}(x)\) 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.
Integration Techniques for Rational Functions
This involves methods to integrate functions expressed as ratios of polynomials or involving roots. Recognizing the form helps decide whether substitution, partial fractions, or trigonometric substitution is appropriate to simplify the integral.
Recommended video:
6:04
Intro to Rational Functions
Trigonometric Substitution
Trigonometric substitution is used to simplify integrals containing expressions like √(x² - a²). By substituting x = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate.
Recommended video:
6:04Introduction to Trigonometric Functions
Domain Considerations and Restrictions
Understanding the domain (x > 1) is crucial because it affects the choice of substitution and the sign of expressions like √(x² - 1). It ensures the substitution is valid and the integral is evaluated correctly within the given constraints.
Recommended video:
5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Textbook Question
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin(θ) sin(2θ) sin(3θ) dθ
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp
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.
∫₋₁³ (4x² - 7) / (2x + 3) dx
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Textbook Question
Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)
To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.
∫ arctan x dx