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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.86
Chapter 8, Problem 8.R.86

82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
86. ∫ (from -∞ to ∞) x³/(1 + x⁸) dx

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1
Identify the type of integral: This is an improper integral with limits from \(-\infty\) to \(\infty\), so we need to consider the behavior of the integrand as \(x\) approaches \(\pm \infty\).
Examine the integrand \(f(x) = \frac{x^3}{1 + x^8}\) for symmetry: Determine if the function is even, odd, or neither by checking \(f(-x)\).
Since \(f(-x) = \frac{(-x)^3}{1 + (-x)^8} = \frac{-x^3}{1 + x^8} = -f(x)\), the function is odd.
Recall that the integral of an odd function over symmetric limits \([-a, a]\) is zero, provided the integral converges.
Check the convergence of the integral by analyzing the behavior of \(f(x)\) as \(x \to \infty\): Since the degree of the denominator is higher than the numerator, the integrand approaches zero fast enough to ensure convergence.

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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 infinite intervals 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 is essential before finding a value.
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Improper Integrals: Infinite Intervals

Even and Odd Functions

A function is odd if f(-x) = -f(x) and even if f(-x) = f(x). For integrals over symmetric intervals [-a, a], the integral of an odd function is zero, while the integral of an even function may be nonzero. Recognizing function symmetry simplifies evaluation.
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Properties of Functions

Behavior of Rational Functions at Infinity

Rational functions are ratios of polynomials. Their behavior as x approaches infinity depends on the degrees of numerator and denominator. If the denominator grows faster, the function approaches zero, which affects the convergence of improper integrals over infinite intervals.
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Intro to Rational Functions
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

109. Average velocity Find the average velocity of a projectile whose velocity over the interval 0 ≤ t ≤ π is given by

v(t) = 10 * sin(3t).

Textbook Question

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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Textbook Question

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

25. ∫ (from -3/2 to -1) dx/(4x² + 12x + 10)

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.

104. About the line y = 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.

57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx