교과서 질문
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (−ln(x)) dx
21
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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 8.8.38
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))
검증된 단계별 안내1
First, identify the integral to be tested for convergence: \(\displaystyle \int_0^{\infty} \frac{d\theta}{\theta^2 - 1}\).
Notice that the integrand has vertical asymptotes where the denominator is zero, i.e., at \(\theta = 1\). This means the integral is improper at \(\theta = 1\) and at the upper limit \(\infty\).
Split the integral into two improper integrals at the point of discontinuity: \(\displaystyle \int_0^1 \frac{d\theta}{\theta^2 - 1} + \int_1^{\infty} \frac{d\theta}{\theta^2 - 1}\).
To test convergence near \(\theta = 1\), analyze the behavior of the integrand as \(\theta \to 1\). For the limit at infinity, compare the integrand to a simpler function such as \(\frac{1}{\theta^2}\) using the Direct Comparison Test or Limit Comparison Test.
Apply the Direct Comparison Test or Limit Comparison Test by computing the limit \(\displaystyle \lim_{\theta \to \infty} \frac{1/(\theta^2 - 1)}{1/\theta^2}\) and check if the integral converges or diverges based on the known behavior of \(\int_1^{\infty} \frac{1}{\theta^2} d\theta\).
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
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate convergence, one must consider limits approaching infinity or points of discontinuity, determining if the integral approaches a finite value.
추천 영상:
11:11
Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test determines convergence by comparing the given integral's integrand to a simpler function with known behavior. If the integrand is smaller than a convergent function or larger than a divergent one, conclusions about convergence or divergence can be drawn.
추천 영상:
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test compares two functions by examining the limit of their ratio as the variable approaches infinity or a point of discontinuity. If the limit is a positive finite number, both integrals either converge or diverge together, aiding in determining the original integral's behavior.
추천 영상:
07:45Limit Comparison Test
관련 실천
교과서 질문
Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx
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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.
∫ x^2 / √(x^2 - 4x + 5) dx
40
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교과서 질문
Evaluate the integrals in Exercises 1–14.
∫ √(y² - 25) / y³ dy, where y > 5
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교과서 질문
Use the substitution u = tan x to evaluate the integral
∫ dx / (1 + sin² x).
38
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교과서 질문
Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)
33
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