Ch. 8 - Integration Techniques

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 8 - Integration Techniques

Problema 8.9.22

Capitolo 8, Problema 8.9.22

7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx

Guida verificata passo dopo passo

First, recognize that the integral is an improper integral because the lower limit is negative infinity: \(\int_{-\infty}^{-2} \frac{1}{x^{2}} \sin\left(\frac{\pi}{2}\right) \, dx\).

Note that \(\sin\left(\frac{\pi}{2}\right)\) is a constant value. Evaluate this constant to simplify the integral.

Rewrite the integral by factoring out the constant \(\sin\left(\frac{\pi}{2}\right)\), so the integral becomes \(\sin\left(\frac{\pi}{2}\right) \int_{-\infty}^{-2} \frac{1}{x^{2}} \, dx\).

Set up the improper integral as a limit: \(\lim_{t \to -\infty} \sin\left(\frac{\pi}{2}\right) \int_{t}^{-2} \frac{1}{x^{2}} \, dx\).

Evaluate the integral \(\int \frac{1}{x^{2}} \, dx\) by rewriting it as \(\int x^{-2} \, dx\) and using the power rule for integration, then apply the limits and take the limit as \(t\) approaches \(-\infty\) to determine if the integral converges or diverges.

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Improper Integrals

Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, limits are used to approach the problematic points, determining if the integral converges to a finite value or diverges.

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Percorso guidato

11:11

Improper Integrals: Infinite Intervals

Behavior of the Integrand

Understanding the integrand's behavior, such as continuity and boundedness, is crucial. In this problem, the integrand includes 1/x² and a constant sine term, so analyzing how 1/x² behaves as x approaches negative infinity helps determine convergence.

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05:22

Completing the Square to Rewrite the Integrand

Evaluation of Definite Integrals with Constants

When the integrand contains constant factors, they can be factored out of the integral to simplify evaluation. Recognizing that sin(π/2) is a constant (equal to 1) allows focusing on integrating 1/x² over the given interval.

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Percorso guidato

05:43

Definition of the Definite Integral

Pratica correlata

Domanda del libro di testo

7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.

18. ∫ dx / (225 − 16x²)

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Domanda del libro di testo

7–84. Evaluate the following integrals.

9. ∫ from 4 to 6 [1 / √(8x – x²)] dx

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Domanda del libro di testo

9–40. Integration by parts Evaluate the following integrals using integration by parts.

14. ∫ s · e⁻²ˢ ds

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Domanda del libro di testo

7–64. Integration review Evaluate the following integrals.