Textbook Question
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
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 8.8.64
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 e to e^e of (ln(ln x) dx)
Verified step by step guidance1
Identify the integral to be tested for convergence: \(\int_{e}^{e^{e}} \ln(\ln x) \, dx\).
Note the interval of integration is finite, from \(x = e\) to \(x = e^{e}\), and the integrand is \(\ln(\ln x)\).
Check the behavior of the integrand on the interval: since \(x \geq e\), \(\ln x \geq 1\), so \(\ln(\ln x)\) is defined and continuous on \([e, e^{e}]\).
Since the integrand is continuous on a closed and bounded interval, the integral is a proper integral and therefore converges.
Conclude that no comparison test is necessary here because the integral is over a finite interval with a continuous integrand, ensuring convergence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Improper Integrals and Convergence
Improper integrals involve integration over unbounded intervals or integrands with unbounded behavior. To determine convergence, we analyze the limit of the integral as the interval approaches infinity or a problematic point. Understanding whether the integral converges or diverges is essential for evaluating integrals like the one given.
Recommended video:
11:11
Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test compares the given integral's integrand to a known function with established convergence behavior. If the integrand is smaller than a convergent function or larger than a divergent function on the interval, we can conclude about the integral's convergence or divergence accordingly.
Recommended video:
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test involves taking the limit of the ratio of the given integrand to a simpler function with known convergence properties. If the limit is a positive finite number, both integrals share the same convergence behavior, making this test useful when direct comparison is difficult.
Recommended video:
07:45Limit Comparison Test
Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
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Textbook Question
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 (dx / √(x² - 1))
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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.
∫ √(9 - w²) dw / w²
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₀^(π/6) √(1 + sin(x)) dx
(Hint: Multiply by √((1 - sin(x)) / (1 - sin(x))))
Textbook Question
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
86. Find the volume of the solid generated by revolving the region about the x-axis.
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