Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
12. ∫ (8x + 5)/(2x² + 3x + 1) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.R.82
82-88. Improper integrals Evaluate the following integrals or show that the integral diverges.
82. ∫ (from -∞ to -1) dx/(x - 1)⁴
Verified step by step guidance1
Identify the type of improper integral: Since the integral has a lower limit of -\(\infty\), this is an improper integral with an infinite limit of integration.
Rewrite the integral as a limit: Express the integral from -\(\infty\) to -1 as \( \lim_{t \to -\infty} \int_{t}^{-1} \frac{1}{(x - 1)^4} \, dx \).
Find the antiderivative: To integrate \( \frac{1}{(x - 1)^4} \), rewrite it as \( (x - 1)^{-4} \) and use the power rule for integration, which states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \).
Evaluate the definite integral: Substitute the antiderivative back into the limit expression and evaluate it at the bounds \( t \) and \( -1 \).
Take the limit as \( t \to -\infty \): Analyze the behavior of the antiderivative as \( t \) approaches negative infinity 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 Rational Functions Near Singularities
Rational functions like 1/(x - a)^n can have vertical asymptotes at x = a. Understanding how the function behaves near these singularities helps determine if the integral converges or diverges, especially when the exponent n affects the rate of growth or decay.
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6:04Intro to Rational Functions
Limit Comparison and Convergence Tests
To decide if an improper integral converges, comparison tests or limit comparison tests are used by comparing the integrand to a simpler function with known integral behavior. This method helps establish convergence or divergence without explicit integration.
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07:45Limit Comparison Test
Related Practice
Textbook Question
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Textbook Question
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c. As a increases, L(a) increases as what power of a?
Textbook Question
2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
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12. ∫ from -5 to 0 of dx / √(4 - x)
Textbook Question
108. Arc length Find the length of the curve y = ln(x) on the interval [1, e^2].