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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 8 - Integration TechniquesProblem 8.R.89
Chapter 8, Problem 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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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Textbook Question

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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