Textbook Question
[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.
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 8.8.62
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 ((1 / ln x) dx)
Verified step by step guidance1
Identify the integral to test for convergence: \(\displaystyle \int_{2}^{\infty} \frac{1}{\ln x} \, dx\).
Recognize that the integral has an infinite upper limit, so we are dealing with an improper integral and need to analyze its behavior as \(x \to \infty\).
Consider comparing the integrand \(\frac{1}{\ln x}\) to a simpler function to determine convergence. Since \(\ln x\) grows without bound but slowly, think about how \(\frac{1}{\ln x}\) compares to functions like \(\frac{1}{x^p}\) or \(\frac{1}{x}\) for large \(x\).
Use the Direct Comparison Test or the Limit Comparison Test by choosing a suitable comparison function. For example, compare \(\frac{1}{\ln x}\) with \(\frac{1}{x}\) because \(\ln x\) grows slower than \(x\), so \(\frac{1}{\ln x}\) decreases slower than \(\frac{1}{x}\).
Evaluate the limit of the ratio of the integrands as \(x \to \infty\): \(\lim_{x \to \infty} \frac{\frac{1}{\ln x}}{\frac{1}{x}} = \lim_{x \to \infty} \frac{x}{\ln x}\). Since this limit goes to infinity, the behavior of \(\frac{1}{\ln x}\) is not dominated by \(\frac{1}{x}\), so use this information to conclude about convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate convergence, we consider limits of definite integrals as the interval approaches infinity or the point of discontinuity. Understanding how to set up and interpret these limits is essential for determining if the integral converges or diverges.
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Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test determines convergence by comparing the given integral to another integral with a known behavior. If the integrand is smaller than a convergent integral or larger than a divergent integral, conclusions about convergence or divergence can be drawn. This test requires identifying suitable comparison functions that bound the original integrand.
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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. If the limit is a positive finite number, both integrals either converge or diverge together. This test is useful when direct comparison is difficult but the asymptotic behavior of the integrand is similar to a simpler function.
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07:45Limit Comparison Test
Related Practice
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos(2x)) dx
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Textbook Question
Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.
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Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ cos^(-1)(√x) / √x dx
Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - x²)^(1/2) / x⁴ dx