Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
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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 8.8.80
In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁴ [x / (x² + 9)^(2/5)] dx
Verified step by step guidance1
Identify the type of improper integral: Since the integral has a lower limit of negative infinity, it is an improper integral due to an infinite limit of integration.
Rewrite the integral as a limit to handle the improper nature: Express the integral as \(\lim_{t \to -\infty} \int_{t}^{4} \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).
Consider the behavior of the integrand as \(x \to -\infty\): Analyze the function \(\frac{x}{(x^{2} + 9)^{2/5}}\) to determine if the integral converges by comparing it to a simpler function whose integral behavior is known.
Find the antiderivative of the integrand: Use substitution methods, such as letting \(u = x^{2} + 9\), to find an expression for the indefinite integral \(\int \frac{x}{(x^{2} + 9)^{2/5}} \, dx\).
Evaluate the definite integral using the antiderivative and then take the limit as \(t \to -\infty\): Substitute the limits into the antiderivative expression and analyze the limit 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, we use limits to approach the problematic points, determining if the integral converges (has a finite value) or diverges (does not).
Recommended video:
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Improper Integrals: Infinite Intervals
Convergence Tests for Improper Integrals
To decide if an improper integral converges, we analyze the behavior of the integrand near infinity or discontinuities. Techniques include comparison tests and evaluating limits of the integral's partial sums to check if they approach a finite number.
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11:11Improper Integrals: Infinite Intervals
Integration Techniques for Rational Functions with Exponents
Integrals involving rational functions with fractional exponents often require substitution or algebraic manipulation. Recognizing patterns and applying appropriate substitutions simplifies the integral, making it easier to evaluate or determine convergence.
Recommended video:
04:09The Power Rule: Negative & Rational Exponents
Related Practice
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 √(2x - x^2) dx
Textbook Question
Solve the initial value problems in Exercises 53–56 for y as a function of x.
√(x² - 9) (dy/dx) = 1, where x > 3, y(5) = ln 3
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.
∫ (sec t + cot t)² dt
Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ e^(-y) cos(y) dy
Textbook Question
In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.
∫ x^2 / √(x^2 - 4x + 5) dx