Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ cos(θ / 2) cos(7θ) dθ
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.3.24
Evaluate the integrals in Exercises 23–32.
∫₀^π √(1 - cos(2x)) dx
Verified step by step guidance1
Recognize that the integral involves the expression \( \sqrt{1 - \cos(2x)} \). Recall the trigonometric identity \( 1 - \cos(2x) = 2\sin^2(x) \). Use this to rewrite the integrand.
Substitute the identity into the integral to get \( \int_0^{\pi} \sqrt{2\sin^2(x)} \, dx \). Since the square root of a square is the absolute value, rewrite the integrand as \( \int_0^{\pi} \sqrt{2} |\sin(x)| \, dx \).
Consider the behavior of \( \sin(x) \) on the interval \( [0, \pi] \). Since \( \sin(x) \) is nonnegative on this interval, \( |\sin(x)| = \sin(x) \). Simplify the integral accordingly.
Factor out the constant \( \sqrt{2} \) from the integral to get \( \sqrt{2} \int_0^{\pi} \sin(x) \, dx \).
Evaluate the integral \( \int_0^{\pi} \sin(x) \, dx \) by finding the antiderivative of \( \sin(x) \), which is \( -\cos(x) \), and then apply the Fundamental Theorem of Calculus by substituting the limits \( 0 \) and \( \pi \).
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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, the identity for cos(2x), such as cos(2x) = 1 - 2sin²(x), helps simplify the integrand √(1 - cos(2x)) into a more manageable form for integration.
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Verifying Trig Equations as Identities
Definite Integration
Definite integration calculates the exact area under a curve between two limits, here from 0 to π. Understanding how to evaluate integrals with specific bounds is essential to find the numerical value of the integral after simplifying the integrand.
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05:43Definition of the Definite Integral
Simplification of Radicals in Integrals
Simplifying expressions under a square root before integrating can make the integral easier to solve. Recognizing that √(1 - cos(2x)) can be rewritten using trigonometric identities reduces complexity and allows the use of standard integral formulas.
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06:13Limits of Rational Functions with Radicals
Related Practice
Textbook Question
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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 2 to ∞ of ((1 / ln x) dx)
Textbook Question
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))
Textbook Question
Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)
To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.
∫ arctan x dx