Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3
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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 8.8.48
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 2 of (dx / (1 - x))
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Identify the integral to be tested for convergence: \(\int_0^2 \frac{dx}{1 - x}\).
Notice that the integrand \(\frac{1}{1 - x}\) has a vertical asymptote at \(x = 1\) because the denominator becomes zero, which lies within the interval of integration \([0, 2]\). This means the integral is an improper integral.
Split the integral at the point of discontinuity to handle it as two improper integrals: \(\int_0^1 \frac{dx}{1 - x} + \int_1^2 \frac{dx}{1 - x}\).
For each integral, rewrite them as limits approaching the point of discontinuity: \(\lim_{t \to 1^-} \int_0^t \frac{dx}{1 - x}\) and \(\lim_{s \to 1^+} \int_s^2 \frac{dx}{1 - x}\).
Evaluate or analyze the behavior of these limits to determine if they converge or diverge. You may use comparison tests or direct integration to assess the convergence of each part.
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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 integrals with infinite limits or integrands with discontinuities within the interval. In this problem, the integrand has a vertical asymptote at x = 1, making the integral improper. Understanding how to handle such integrals by taking limits is essential to determine convergence.
Recommended video:
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Improper Integrals: Infinite Intervals
Direct Comparison Test
The Direct Comparison Test compares the given integrand to a simpler function whose convergence behavior is known. If the integrand is smaller than a convergent function or larger than a divergent function on the interval, conclusions about convergence or divergence can be drawn.
Recommended video:
09:25Direct Comparison Test
Limit Comparison Test
The Limit Comparison Test involves taking the limit of the ratio of the given integrand to a known benchmark function as the variable approaches the problematic point. If the limit is a positive finite number, both integrals share the same convergence behavior, aiding in determining the original integral's convergence.
Recommended video:
07:45Limit Comparison Test
Related Practice
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.
∫ dx / (x - √x)
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Textbook Question
Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.
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Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) dx
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 0 to π/2 of (cot θ dθ)
Textbook Question
Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt
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