Domanda del libro di testo
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞^∞ (x dx) / (x² + 4)^(3/2)
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Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Capitolo 8, Problema 8.8.22
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
Guida verificata passo dopo passo1
Recognize that the integral is of the form \(\int_0^{\infty} e^{-a\theta} \sin(b\theta) \, d\theta\) where \(a = 1\) and \(b = 1\), but here the integrand is \(2 e^{-\theta} \sin \theta\). Factor out the constant 2 to write the integral as \(2 \int_0^{\infty} e^{-\theta} \sin \theta \, d\theta\).
Recall the standard formula for the integral \(\int_0^{\infty} e^{-p x} \sin(q x) \, dx = \frac{q}{p^2 + q^2}\), valid for \(p > 0\). Here, identify \(p = 1\) and \(q = 1\).
Apply the formula to evaluate \(\int_0^{\infty} e^{-\theta} \sin \theta \, d\theta = \frac{1}{1^2 + 1^2} = \frac{1}{2}\).
Multiply the result by the constant factor 2 that was factored out initially, so the original integral becomes \(2 \times \frac{1}{2}\).
Conclude that the value of the integral is the product found in the previous step, which completes the evaluation without using tables.
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Improper Integrals
Improper integrals involve integration over an infinite interval or integrands with infinite discontinuities. To evaluate them, one typically takes limits to handle the infinite bounds, ensuring the integral converges to a finite value.
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Improper Integrals: Infinite Intervals
Integration of Exponential and Trigonometric Functions
Integrals involving products of exponential and trigonometric functions can be solved using integration techniques such as integration by parts or recognizing standard integral forms. These often result in expressions involving both sine and cosine terms.
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Convergence of Integrals with Oscillatory Functions
When integrating functions like e^(−θ) sinθ over [0, ∞), the exponential decay ensures the integral converges despite the oscillatory sine term. Understanding how the decay dominates oscillations is key to confirming convergence.
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Percorso guidato
07:51Choosing a Convergence Test
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