Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ 1 / (x√x + 9) dx
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.22
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
Verified step by step guidance1
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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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, 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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05:11Integrals of General Exponential Functions
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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07:51Choosing a Convergence Test
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
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Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt
Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from -∞ to ∞ of ((dx) / (e^x + e^(-x)))
1
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