Domanda del libro di testo
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) 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 8.8.66
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 1 to ∞ of ((1 / (e^x - 2^x)) dx)
Guida verificata passo dopo passo1
First, identify the integral to be tested for convergence: \(\displaystyle \int_1^{\infty} \frac{1}{e^x - 2^x} \, dx\).
Examine the behavior of the integrand as \(x \to \infty\). Since both \(e^x\) and \$2^x$ grow exponentially, compare their growth rates to understand the dominant term in the denominator.
Determine which function grows faster: \(e^x\) or \$2^x$. Recall that \(e \approx 2.718\), so $e^x$ grows faster than \$2^x$. Therefore, for large $x\(, \)e^x - 2^x$ behaves approximately like $e^x$.
Use the Direct Comparison Test by comparing the integrand to a simpler function that behaves like \(\frac{1}{e^x}\) for large \(x\). Since \(\frac{1}{e^x}\) is a convergent integral on \([1, \infty)\), this comparison can help determine convergence.
Alternatively, apply the Limit Comparison Test by computing the limit \(\displaystyle \lim_{x \to \infty} \frac{\frac{1}{e^x - 2^x}}{\frac{1}{e^x}}\) to confirm if the integrals behave similarly at infinity, which will indicate whether the original integral converges or diverges.
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Improper Integrals and Convergence
Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To determine convergence, we evaluate the limit of the integral as the upper bound approaches infinity. If this limit exists and is finite, the integral converges; otherwise, it diverges.
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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 simpler function whose convergence behavior is known. If the integrand is less than or equal to a convergent function, the integral converges; if it is greater than or equal to a divergent function, the integral diverges.
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Percorso guidato
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test uses the limit of the ratio of the given integrand to a known function as x approaches infinity. If the limit is a positive finite number, both integrals either converge or diverge together, helping to determine the behavior of complex integrals.
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Percorso guidato
07:45Limit Comparison Test
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Domanda del libro di testo
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 π/2 of (cot θ dθ)
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Domanda del libro di testo
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
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Domanda del libro di testo
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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Domanda del libro di testo
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
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Domanda del libro di testo
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
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