Textbook Question
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (x² √(x² + 1))
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 8.3.6
Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) dx
Verified step by step guidance1
Recognize that the integral involves a power of cosine: \(\int \cos^{3}(4x) \, dx\). To simplify, use the identity for odd powers of cosine: express \(\cos^{3}(\theta)\) as \(\cos(\theta) \cdot \cos^{2}(\theta)\).
Rewrite \(\cos^{2}(\theta)\) using the Pythagorean identity: \(\cos^{2}(\theta) = 1 - \sin^{2}(\theta)\). So, \(\cos^{3}(4x) = \cos(4x) \cdot (1 - \sin^{2}(4x))\).
Substitute this back into the integral: \(\int \cos(4x) (1 - \sin^{2}(4x)) \, dx = \int \cos(4x) \, dx - \int \cos(4x) \sin^{2}(4x) \, dx\).
Use substitution for the second integral: let \(u = \sin(4x)\), then \(du = 4 \cos(4x) \, dx\), or equivalently, \(\cos(4x) \, dx = \frac{du}{4}\). Rewrite the integral in terms of \(u\).
Integrate the resulting expression in \(u\), then substitute back \(u = \sin(4x)\) to express the answer in terms of \(x\). Don't forget to add the constant of integration \(C\).
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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. For integrals like ∫ cos³(4x) dx, identities such as the power-reduction or product-to-sum formulas help rewrite powers of cosine into expressions easier to integrate.
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Integration of Powers of Trigonometric Functions
Integrating powers of sine or cosine often requires reducing the power using identities or substitution. For odd powers, one factor is separated, and the remaining even power is converted using identities, simplifying the integral into basic trigonometric integrals.
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Substitution Method
The substitution method involves changing variables to simplify an integral. For example, when integrating functions like cos³(4x), substituting u = 4x transforms the integral into a simpler form, making it easier to apply standard integration techniques.
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Related Practice
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Textbook Question
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∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx