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 8.3.32
Evaluate the integrals in Exercises 23–32.
∫₋π^π (1 - cos²(t))^(3/2) dt
Verified step by step guidance1
Recognize that the integrand is \( (1 - \cos^{2}(t))^{3/2} \). Recall the Pythagorean identity \( \sin^{2}(t) + \cos^{2}(t) = 1 \), which allows us to rewrite the integrand in terms of \( \sin(t) \).
Rewrite the integrand as \( (\sin^{2}(t))^{3/2} = |\sin(t)|^{3} \) because \( (\sin^{2}(t))^{3/2} = (\sin^{2}(t))^{1 \cdot 3/2} = |\sin(t)|^{3} \).
Since the integral is from \( -\pi \) to \( \pi \), and \( |\sin(t)|^{3} \) is an even function (because \( |\sin(-t)| = |\sin(t)| \)), use the property of even functions to simplify the integral: \( \int_{-\pi}^{\pi} |\sin(t)|^{3} dt = 2 \int_{0}^{\pi} |\sin(t)|^{3} dt \).
On the interval \( [0, \pi] \), \( \sin(t) \) is non-negative, so \( |\sin(t)|^{3} = \sin^{3}(t) \). Thus, the integral becomes \( 2 \int_{0}^{\pi} \sin^{3}(t) dt \).
To evaluate \( \int \sin^{3}(t) dt \), use the reduction formula or rewrite \( \sin^{3}(t) \) as \( \sin(t) \cdot \sin^{2}(t) = \sin(t)(1 - \cos^{2}(t)) \), then use substitution \( u = \cos(t) \) to integrate.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing that 1 - cos²(t) equals sin²(t) simplifies the integrand, making the integral easier to evaluate.
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Definite Integrals and Symmetry
Definite integrals calculate the net area under a curve between two limits. When the integrand is an even or odd function, symmetry properties can simplify the integral, such as doubling the integral from 0 to π if the function is even.
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05:43Definition of the Definite Integral
Power Functions and Integration Techniques
Integrating functions raised to a power often requires substitution or recognizing standard integral forms. Here, raising sin(t) to the third power involves understanding how to integrate powers of sine, possibly using reduction formulas or substitution.
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07:32Representing Functions as Power Series
Related Practice
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫ 7cos⁷(t) dt
Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ 1/(x(ln(x))²) dx
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
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))