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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.86
Capitolo 8, Problema 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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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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11:11
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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06:21
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
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