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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 8.9.22
Chapter 8, Problem 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

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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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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 them, limits are used to approach the problematic points, determining if the integral converges to a finite value or diverges.
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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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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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Definition of the Definite Integral
Related Practice
Textbook Question

72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].

Textbook Question

60–69. Completing the square Evaluate the following integrals.

65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)

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Textbook Question

67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:

sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]

sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]

cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]

68. ∫ sin(5x)sin(7x) dx

Textbook Question

4. Is a reduction formula an analytical method or a numerical method? Explain.

Textbook Question

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.

8. ∫ sin 3x cos 2x dx

Textbook Question

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

29. ∫ e⁻ˣ sin(4x) dx