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Ch. 8 - Techniques of Integration
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 8 - Techniques of IntegrationProblema 8.PE.67
Capitolo 8, Problema 8.PE.67

Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx

Guida verificata passo dopo passo
1
First, recognize that the integral is improper because the limits of integration are from \(-\infty\) to \(\infty\).
Rewrite the integrand \(\frac{2}{e^x + e^{-x}}\) in a simpler form. Notice that \(e^x + e^{-x} = 2 \cosh x\), so the integrand becomes \(\frac{2}{2 \cosh x} = \frac{1}{\cosh x}\).
Express the integral as \(\int_{-\infty}^{\infty} \frac{1}{\cosh x} \, dx\). This is a standard integral involving the hyperbolic secant function, since \(\frac{1}{\cosh x} = \operatorname{sech} x\).
To determine convergence, analyze the behavior of \(\operatorname{sech} x\) as \(x \to \pm \infty\). Since \(\cosh x\) grows exponentially, \(\operatorname{sech} x\) approaches zero rapidly, suggesting the integral may converge.
Set up the integral as the sum of two limits: \(\lim_{a \to -\infty} \int_a^0 \operatorname{sech} x \, dx + \lim_{b \to \infty} \int_0^b \operatorname{sech} x \, dx\). Evaluate these integrals using the antiderivative of \(\operatorname{sech} x\), which is \(2 \arctan(\tanh(\frac{x}{2}))\), to check for convergence.

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Concetti chiave

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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 approach the problematic points, determining if the integral converges (has a finite value) or diverges (does not).
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Improper Integrals: Infinite Intervals

Convergence and Divergence of Integrals

An integral converges if its limit exists and is finite; otherwise, it diverges. For integrals over infinite intervals, convergence depends on the behavior of the integrand as the variable approaches infinity or negative infinity.
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Divergence Test (nth Term Test)

Behavior of Exponential Functions

Exponential functions like e^x and e^(-x) grow or decay rapidly. Understanding their limits as x approaches ±∞ helps analyze the integrand's behavior, which is crucial for determining the convergence of the integral.
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Graphs of Exponential Functions
Pratica correlata
Domanda del libro di testo

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

133. ∫ (sin²x) / (1 + sin²x) dx

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Domanda del libro di testo

A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of

f(x) = √(1 + x⁴)

lies between 0 and 8.

Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1

with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?

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Domanda del libro di testo

Evaluate the improper integrals in Exercises 53–62.

∫ from 3 to ∞ of (2 / (u² − 2u)) du

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Domanda del libro di testo

Evaluate the improper integrals in Exercises 53–62.

∫ from 0 to 3 of (1 / √(9 − x²)) dx

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Domanda del libro di testo

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

127. ∫ (ln x) / (x + x ln x) dx

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Domanda del libro di testo

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ x dx / √(2 − x)

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