Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy
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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 8.5.52
Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ
Verified step by step guidance1
Start by rewriting the integral \( \int \frac{1}{\cos \theta + \sin 2\theta} \, d\theta \) and recall the double-angle identity for sine: \( \sin 2\theta = 2 \sin \theta \cos \theta \). Substitute this into the integral to get \( \int \frac{1}{\cos \theta + 2 \sin \theta \cos \theta} \, d\theta \).
Factor \( \cos \theta \) from the denominator: \( \cos \theta + 2 \sin \theta \cos \theta = \cos \theta (1 + 2 \sin \theta) \). So the integral becomes \( \int \frac{1}{\cos \theta (1 + 2 \sin \theta)} \, d\theta \).
Rewrite the integral as \( \int \frac{1}{\cos \theta (1 + 2 \sin \theta)} \, d\theta = \int \frac{\sec \theta}{1 + 2 \sin \theta} \, d\theta \), since \( \sec \theta = \frac{1}{\cos \theta} \).
Use the substitution \( u = 1 + 2 \sin \theta \). Then, compute \( du = 2 \cos \theta \, d\theta \), which implies \( d\theta = \frac{du}{2 \cos \theta} \).
Rewrite the integral in terms of \( u \) and \( \theta \), and express \( \sec \theta \, d\theta \) using the substitution. This will allow you to simplify the integral and proceed with integration techniques such as partial fractions or further substitutions.
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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 of the variables. They are essential for simplifying expressions like cos θ + sin 2θ, where sin 2θ can be rewritten using the double-angle identity sin 2θ = 2 sin θ cos θ to facilitate integration.
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Verifying Trig Equations as Identities
Integration of Rational Trigonometric Functions
Integrating rational functions involving trigonometric expressions often requires algebraic manipulation or substitution. Recognizing patterns or rewriting the integrand in terms of a single trigonometric function can simplify the integral and make standard integration techniques applicable.
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Substitution Method in Integration
The substitution method involves changing variables to simplify an integral. For trigonometric integrals, substituting expressions like u = sin θ or u = cos θ can transform the integral into a more manageable form, allowing the use of basic integral formulas.
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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
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
Textbook Question
Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.
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