교과서 질문
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx
14
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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 8.8.30
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
검증된 단계별 안내1
Identify the integral to solve: \(\int_{2}^{4} \frac{dt}{t \sqrt{t^{2} - 4}}\).
Recognize that the integrand contains a square root of the form \(\sqrt{t^{2} - a^{2}}\), suggesting a trigonometric substitution. Use the substitution \(t = 2 \sec(\theta)\), where \(a = 2\).
Compute the differential \(dt\) in terms of \(d\theta\): since \(t = 2 \sec(\theta)\), then \(dt = 2 \sec(\theta) \tan(\theta) d\theta\).
Rewrite the integral in terms of \(\theta\) by substituting \(t\), \(dt\), and simplifying the expression under the square root: \(\sqrt{t^{2} - 4} = \sqrt{4 \sec^{2}(\theta) - 4} = 2 \tan(\theta)\).
Change the limits of integration from \(t\) to \(\theta\) using \(t = 2 \sec(\theta)\), then simplify the integral and integrate with respect to \(\theta\).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Improper Integrals and Convergence
Understanding when an integral converges is crucial, especially for integrals with infinite limits or integrands with singularities. In this problem, the integral is definite and the integrand is continuous on [2,4], ensuring convergence. Recognizing convergence allows safe evaluation without concern for divergence.
추천 영상:
가이드 코스
11:11
Improper Integrals: Infinite Intervals
Trigonometric Substitution
Trigonometric substitution is a technique used to simplify integrals involving expressions like √(t² − a²). By substituting t = a sec(θ), the radical simplifies using trigonometric identities, making the integral easier to evaluate. This method transforms the integral into a trigonometric integral that can be integrated directly.
추천 영상:
가이드 코스
6:04Introduction to Trigonometric Functions
Integration Techniques for Rational Functions
Integrals involving rational functions combined with radicals often require algebraic manipulation or substitution to simplify. Recognizing the structure of the integrand, such as t in the denominator and a radical in the denominator, guides the choice of substitution and integration steps to find an explicit antiderivative.
추천 영상:
6:04Intro to Rational Functions
관련 실천
교과서 질문
Evaluate the integrals in Exercises 1–14.
∫ (3 dx) / √(1 + 9x²)
27
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교과서 질문
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ t² e^(4t) dt
42
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교과서 질문
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ dt / (tan(t)√4 - sin^2(t))
21
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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 2 to ∞ of ((x dx) / √(x⁴ - 1))
교과서 질문
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 16x^3 (ln(x))^2 dx
17
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