Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (1 - r²)^(5/2) / r⁸ dr
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 8.8.54
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 0 to ∞ of (dθ / (1 + e^θ))
Verified step by step guidance1
Identify the integral to be tested for convergence: \(\displaystyle \int_0^{\infty} \frac{d\theta}{1 + e^{\theta}}\).
Examine the behavior of the integrand \(f(\theta) = \frac{1}{1 + e^{\theta}}\) as \(\theta \to \infty\) and as \(\theta \to 0\) to understand the nature of the integral over the interval \([0, \infty)\).
For large \(\theta\), note that \(e^{\theta}\) grows very quickly, so \(f(\theta)\) behaves like \(\frac{1}{e^{\theta}} = e^{-\theta}\). This suggests comparing the integrand to \(g(\theta) = e^{-\theta}\), a function with a known convergent integral over \([0, \infty)\).
Apply the Direct Comparison Test by checking if \(0 \leq f(\theta) \leq g(\theta)\) for sufficiently large \(\theta\). Since \(\frac{1}{1 + e^{\theta}} < \frac{1}{e^{\theta}}\) for \(\theta > 0\), the integral of \(f(\theta)\) is bounded above by the integral of \(g(\theta)\), which converges.
Conclude that since \(\int_0^{\infty} e^{-\theta} d\theta\) converges and \(f(\theta)\) is less than or equal to \(g(\theta)\) for large \(\theta\), by the Direct Comparison Test, the original integral \(\int_0^{\infty} \frac{d\theta}{1 + e^{\theta}}\) also converges.
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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 convergence, we consider limits of definite integrals as the bounds approach infinity or points of discontinuity. Understanding this helps determine if the integral converges to a finite value or diverges.
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Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test compares the given integral's integrand to a simpler function whose convergence behavior is known. If the integrand is less than or equal to a convergent function, the integral converges; if it is greater than or equal to a divergent function, it diverges. This test requires finding appropriate bounding functions.
Recommended video:
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test uses the limit of the ratio of two functions as the variable approaches infinity. If the limit is a positive finite number, both integrals either converge or diverge together. This test is useful when direct comparison is difficult but the integrand behaves similarly to a known function at infinity.
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07:45Limit Comparison Test
Related Practice
Textbook Question
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz
Textbook Question
Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx
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Textbook Question
Finding arc length
Find the length of the curve
y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ 3 sinh(x/2 + ln 5) dx
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dt / t√(3 + t²)
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