Textbook Question
Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx
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.PE.8
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x sin(x) cos(x) dx
Verified step by step guidance1
Recognize that the integral involves the product of functions: \(x\) and \(\sin(x) \cos(x)\). Integration by parts is suitable here, where we let one part be differentiated and the other integrated.
Use the trigonometric identity to simplify the integrand: \(\sin(x) \cos(x) = \frac{1}{2} \sin(2x)\). So the integral becomes \(\int x \cdot \frac{1}{2} \sin(2x) \, dx = \frac{1}{2} \int x \sin(2x) \, dx\).
Set up integration by parts with \(u = x\) (which simplifies upon differentiation) and \(dv = \sin(2x) \, dx\) (which can be integrated easily). Then compute \(du = dx\) and find \(v\) by integrating \(dv\).
Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\). Substitute the expressions for \(u\), \(v\), \(du\), and \(dv\) into this formula.
Evaluate the resulting integral \(\int v \, du\) and simplify the expression to write the integral in terms of elementary functions plus the constant of integration.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration by Parts
Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the problem.
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Integration by Parts for Definite Integrals
Trigonometric Identities
Trigonometric identities, such as sin(2x) = 2 sin(x) cos(x), help simplify integrals involving products of sine and cosine. Using these identities can reduce complex expressions into more manageable forms for integration.
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Integration of Basic Trigonometric Functions
Knowing how to integrate basic trigonometric functions like sin(x) and cos(x) is essential. These integrals often appear after applying integration by parts or simplifying expressions, enabling the completion of the integration process.
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Related Practice
Textbook Question
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 1 to ∞ of ((ln z) / z) dz
Textbook Question
Evaluate the integrals in Exercises 37–44.
∫ sec²(θ) sin³(θ) dθ
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sin(2θ) dθ / (1 + cos(2θ))²
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
131. ∫ dx / (x√(1 − x⁴))
Textbook Question
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx
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