Domanda del libro di testo
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv
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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.PE.65
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 1 to ∞ of ((ln z) / z) dz
Guida verificata passo dopo passo1
Identify the integral to analyze: \(\displaystyle \int_1^{\infty} \frac{\ln z}{z} \, dz\).
Recognize that this is an improper integral because the upper limit of integration is infinite.
Consider the behavior of the integrand \(\frac{\ln z}{z}\) as \(z \to \infty\). To determine convergence, analyze the limit of the integral or use a comparison test.
Use substitution to simplify the integral: let \(t = \ln z\), which implies \(z = e^t\) and $dz = e^t dt$. Rewrite the integral in terms of \(t\).
After substitution, express the integral with new limits and integrand, then analyze whether the resulting integral converges or diverges by evaluating the limit as \(t \to \infty\).
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Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, limits are used to define the integral as a limit of definite integrals over finite intervals. Determining convergence or divergence depends on whether this limit exists and is finite.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Convergence and Divergence of Integrals
An improper integral converges if the limit defining it exists and is finite; otherwise, it diverges. Testing convergence often involves comparison tests or evaluating the behavior of the integrand as the variable approaches infinity or a point of discontinuity.
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Percorso guidato
05:44Divergence Test (nth Term Test)
Behavior of Logarithmic Functions in Integrals
Logarithmic functions like ln(z) grow slowly as z approaches infinity. When combined with other functions, such as 1/z, their growth rate affects the convergence of the integral. Understanding how ln(z)/z behaves for large z is key to determining if the integral converges.
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5:26Graphs of Logarithmic Functions
Pratica correlata
Domanda del libro di testo
You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)
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Domanda del libro di testo
Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 37–44.
∫ sec²(θ) sin³(θ) dθ
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Domanda del libro di testo
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x sin(x) cos(x) dx
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Domanda del libro di testo
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (e^x + e^(3x)) / e^(2x) dx
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