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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.R.89
Capitolo 8, Problema 8.R.89

89–91. Comparison Test Determine whether the following integrals converge or diverge.
89. ∫ (from 1 to ∞) dx/(x⁵ + x⁴ + x³ + 1)

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Identify the integral to analyze: \(\int_1^{\infty} \frac{dx}{x^5 + x^4 + x^3 + 1}\).
To apply the Comparison Test, find a simpler function to compare with the integrand. For large \(x\), the term \(x^5\) dominates the denominator, so consider comparing with \(\frac{1}{x^5}\).
Check if \(\frac{1}{x^5 + x^4 + x^3 + 1} \leq \frac{1}{x^5}\) for \(x \geq 1\). Since \(x^5 + x^4 + x^3 + 1 \geq x^5\), this inequality holds.
Recall that the integral \(\int_1^{\infty} \frac{1}{x^5} dx\) converges because the exponent 5 is greater than 1.
By the Comparison Test, since \(\int_1^{\infty} \frac{1}{x^5} dx\) converges and \(\frac{1}{x^5 + x^4 + x^3 + 1} \leq \frac{1}{x^5}\), the original integral also converges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate convergence, one considers the limit of the integral as the bound approaches infinity. Understanding this concept is essential for determining whether the integral converges or diverges.
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Improper Integrals: Infinite Intervals

Comparison Test for Improper Integrals

The Comparison Test helps determine convergence by comparing the given integral to a simpler integral with known behavior. If the integrand is less than or equal to a convergent integral's integrand, the original integral converges; if it is greater than or equal to a divergent integral's integrand, it diverges.
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Percorso guidato
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Improper Integrals: Infinite Intervals

Behavior of Rational Functions at Infinity

For large values of x, the dominant terms in the numerator and denominator dictate the integrand's behavior. Simplifying the integrand by focusing on highest-degree terms helps estimate its decay rate, which is crucial for applying the Comparison Test and assessing convergence.
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Intro to Rational Functions
Pratica correlata
Domanda del libro di testo

101. Comparing volumes Let R be the region bounded by the graph of y = sin(x) and the x-axis on the interval [0, π]. Which is greater, the volume of the solid generated when R is revolved about the x-axis or about the y-axis?

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106. Arc length Find the length of the curve y = (x / 2) * sqrt(3 - x^2) + (3 / 2) * sin^(-1)(x / sqrt(3)) from x = 0 to x = 1.

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Domanda del libro di testo

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

74. ∫ dx/√(√(1 + √x))

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125. Wallis products Complete the following steps to prove a well-known formula discovered by the 17th-century English mathematician John Wallis.

a. Use a reduction formula to show that ∫ from 0 to π of (sin^m x) dx = (m − 1)/m × ∫ from 0 to π of (sin^(m−2) x) dx, for any integer m ≥ 2.

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Domanda del libro di testo

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

68. ∫ (from -1 to 1) dx/(x² + 2x + 5)

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102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

102. About the y-axis

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