Textbook Question
Arc length:
Find the length of the curve y = x², 0 ≤ x ≤ √3/2.
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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 8.4.12
Evaluate the integrals in Exercises 1–14.
∫ √(y² - 25) / y³ dy, where y > 5
Verified step by step guidance1
Identify the integral to solve: \(\int \frac{\sqrt{y^{2} - 25}}{y^{3}} \, dy\), with the condition \(y > 5\).
Recognize that the integrand contains a square root of the form \(\sqrt{y^{2} - a^{2}}\), which suggests using a trigonometric substitution. Since \(y > 5\), set \(y = 5 \sec(\theta)\), where \(\theta\) is in the appropriate domain to keep \(y > 5\).
Compute the differential \(dy\) in terms of \(d\theta\): \(dy = 5 \sec(\theta) \tan(\theta) \, d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(y = 5 \sec(\theta)\) and \(dy\) as above. Simplify the expression inside the square root and the powers of \(y\) accordingly.
Simplify the resulting integral using trigonometric identities, then integrate with respect to \(\theta\). After integration, substitute back \(\theta = \sec^{-1}(\frac{y}{5})\) to express the answer in terms of \(y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Rational Functions Involving Radicals
This concept involves integrating functions where the integrand contains radicals combined with rational expressions. Techniques often include algebraic manipulation or substitution to simplify the integrand into a more manageable form for integration.
Recommended video:
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Integrals Involving Natural Logs: Substitution
Trigonometric Substitution
Trigonometric substitution is a method used to evaluate integrals containing expressions like √(y² - a²). By substituting y = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to solve.
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6:04Introduction to Trigonometric Functions
Definite Domain Considerations and Variable Restrictions
When integrating functions with radicals, the domain restrictions (e.g., y > 5) ensure the expression under the square root is non-negative. Recognizing these constraints is essential to choose the correct substitution and interpret the integral's result properly.
Recommended video:
05:43Definition of the Definite Integral
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ 4x sec²(2x) dx
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫(from 1 to e) x³ ln(x) dx
Textbook Question
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₁^∞ (1 / (x² + 3x)) dx
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Textbook Question
Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)
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