교과서 질문
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ 4x sec²(2x) dx
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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
8장, 문제 8.4.12
Evaluate the integrals in Exercises 1–14.
∫ √(y² - 25) / y³ dy, where y > 5
검증된 단계별 안내1
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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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
07:01
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.
추천 영상:
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.
추천 영상:
05:43Definition of the Definite Integral
관련 실천
교과서 질문
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫(from 1 to e) x³ ln(x) dx
32
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교과서 질문
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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교과서 질문
Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).
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교과서 질문
Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)
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교과서 질문
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from 0 to ∞ of (dθ / (θ² - 1))
32
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