Domanda del libro di testo
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) 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.PE.57
Evaluate the improper integrals in Exercises 53–62.
∫ from 3 to ∞ of (2 / (u² − 2u)) du
Guida verificata passo dopo passo1
First, recognize that the integral is improper because the upper limit is infinity. So, rewrite the integral as a limit: \(\displaystyle \int_{3}^{\infty} \frac{2}{u^{2} - 2u} \, du = \lim_{t \to \infty} \int_{3}^{t} \frac{2}{u^{2} - 2u} \, du\).
Next, factor the denominator to simplify the integrand: \(u^{2} - 2u = u(u - 2)\). So the integrand becomes \(\frac{2}{u(u - 2)}\).
Use partial fraction decomposition to express \(\frac{2}{u(u - 2)}\) as \(\frac{A}{u} + \frac{B}{u - 2}\). Set up the equation \$2 = A(u - 2) + Bu$ and solve for constants $A$ and $B$.
After finding \(A\) and \(B\), rewrite the integral as the sum of two simpler integrals: \(\int \frac{A}{u} \, du + \int \frac{B}{u - 2} \, du\). Integrate each term separately to get logarithmic functions.
Finally, substitute the limits from 3 to \(t\), then take the limit as \(t\) approaches infinity. Evaluate the resulting expression to determine if the improper integral converges or diverges.
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Improper Integrals
Improper integrals involve integration over an infinite interval or where the integrand has an infinite discontinuity. To evaluate them, the integral is expressed as a limit, such as taking the limit as the upper bound approaches infinity. This approach ensures the integral converges to a finite value or diverges.
Video consigliato:
Percorso guidato
11:11
Improper Integrals: Infinite Intervals
Partial Fraction Decomposition
Partial fraction decomposition breaks a rational function into simpler fractions that are easier to integrate. For example, a function like 2/(u² − 2u) can be rewritten as a sum of fractions with linear denominators, facilitating straightforward integration of each term.
Video consigliato:
Percorso guidato
10:07Partial Fraction Decomposition: Distinct Linear Factors
Limits and Convergence of Integrals
Evaluating improper integrals requires analyzing the limit of the integral as the variable approaches infinity. Determining whether this limit exists (converges) or not (diverges) is crucial. Convergence means the area under the curve is finite, allowing the integral to be assigned a value.
Video consigliato:
Percorso guidato
07:51Choosing a Convergence Test
Pratica correlata
Domanda del libro di testo
A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?
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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.
131. ∫ dx / (x√(1 − x⁴))
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Domanda del libro di testo
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(4x) / (x³ + 4x)] dx
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Domanda del libro di testo
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 3 of (1 / √(9 − x²)) dx
27
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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.
127. ∫ (ln x) / (x + x ln x) dx
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