Domanda del libro di testo
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ 1 / √(x² - 2x + 5) dx
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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
Capitolo 8, Problema 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)
Guida verificata passo dopo passo1
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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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.
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Percorso guidato
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.
Video consigliato:
Percorso guidato
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.
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Percorso guidato
07:45Limit Comparison Test
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Domanda del libro di testo
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Domanda del libro di testo
Evaluate the integrals in Exercises 23–32.
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Domanda del libro di testo
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
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