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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.9.31
Capitolo 8, Problema 8.9.31

7–58. Improper integrals Evaluate the following integrals or state that they diverge.
31. ∫ (from 1 to ∞) 1/[v(v + 1)] dv

Guida verificata passo dopo passo
1
Identify that the integral \( \int_1^{\infty} \frac{1}{v(v+1)} \, dv \) is an improper integral because the upper limit is infinite.
Rewrite the integral as a limit: \( \lim_{t \to \infty} \int_1^{t} \frac{1}{v(v+1)} \, dv \). This allows us to handle the infinite limit properly.
Use partial fraction decomposition to simplify the integrand: express \( \frac{1}{v(v+1)} \) as \( \frac{A}{v} + \frac{B}{v+1} \) and solve for constants \( A \) and \( B \).
Integrate the decomposed expression term-by-term over \( v \) from 1 to \( t \), resulting in logarithmic functions.
Evaluate the definite integral by substituting the limits 1 and \( t \), then take the limit as \( t \to \infty \) to determine if the integral converges or diverges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, the integral is expressed as a limit where the bound approaches infinity or the point of discontinuity. Convergence or divergence depends on whether this limit exists finitely.
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Percorso guidato
11:11
Improper Integrals: Infinite Intervals

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational function into simpler fractions that are easier to integrate. For example, 1/[v(v + 1)] can be rewritten as A/v + B/(v + 1), allowing straightforward integration of each term separately.
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10:07
Partial Fraction Decomposition: Distinct Linear Factors

Evaluating Limits of Integrals

When dealing with improper integrals, after integrating the function, you must evaluate the limit of the antiderivative as the variable approaches infinity. This step determines if the integral converges to a finite value or diverges.
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One-Sided Limits
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Domanda del libro di testo

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

53. ∫ (from 0 to 1) ln x dx

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Domanda del libro di testo

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.

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Domanda del libro di testo

59. Perpetual Annuity

Imagine that today you deposit $B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of $I per year from the account forever. The amount of money that must be deposited is:

B = I × ∫(from 0 to ∞) e^(-rt) dt

where r = p/100.

Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of \$5000 per year. How much must you deposit today?

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Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

47. ∫ dx / (x⁻¹ + 1)

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Domanda del libro di testo

71-74. Deriving formulas Evaluate the following integrals. Assume a and b are real numbers and n is a positive integer.

74. ∫xⁿ arcsin(x) dx (Hint: integration by parts.)

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Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.

62. ∫ (-x⁵ - x⁴ - 2x³ + 4x + 3) / (x² + x + 1) dx

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