Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
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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 8.3.34
Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx
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Recall the trigonometric identity: \(\tan^2(x) = \sec^2(x) - 1\). This allows us to rewrite the integral in terms of secant functions.
Rewrite the integral as \(\int \sec(x) \tan^2(x) \, dx = \int \sec(x) (\sec^2(x) - 1) \, dx\).
Distribute \(\sec(x)\) inside the integral to get \(\int (\sec^3(x) - \sec(x)) \, dx\).
Split the integral into two separate integrals: \(\int \sec^3(x) \, dx - \int \sec(x) \, dx\).
Use known integration techniques: recall that \(\int \sec(x) \, dx\) is a standard integral, and for \(\int \sec^3(x) \, dx\), use integration by parts or a reduction formula to express it in terms of \(\int \sec(x) \, dx\).
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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 like tan²(x) = sec²(x) - 1 help simplify integrals involving trigonometric functions. Recognizing and applying these identities can transform complex expressions into more manageable forms for integration.
Recommended video:
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Verifying Trig Equations as Identities
Integration Techniques for Trigonometric Functions
Integrating functions involving sec(x) and tan(x) often requires substitution or rewriting the integrand using identities. Familiarity with standard integrals such as ∫ sec(x) dx and ∫ sec(x) tan(x) dx is essential for solving these problems efficiently.
Recommended video:
6:04Introduction to Trigonometric Functions
Substitution Method
The substitution method involves choosing a part of the integrand as a new variable to simplify the integral. For example, setting u = sec(x) or u = tan(x) can reduce the integral to a basic form, making it easier to evaluate.
Recommended video:
07:33Euler's Method
Related Practice
Textbook Question
Evaluate the integrals in Exercises 39–54.
∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
Textbook Question
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e
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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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))
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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 π to ∞ of ((1 + sin x) / x² dx)
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