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.
∫ (tan θ + 3 / sin θ) 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 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 8.3.30
Evaluate the integrals in Exercises 23–32.
∫_{π/2}^{3π/4} √(1 - sin(2x)) dx
Verified step by step guidance1
Start by examining the integrand: \(\sqrt{1 - \sin(2x)}\). Recognize that the expression inside the square root can be simplified using trigonometric identities.
Recall the double-angle identity for sine: \(\sin(2x) = 2\sin x \cos x\). However, a more useful identity here is the one involving cosine of a double angle: \(1 - \sin(2x) = 1 - 2\sin x \cos x\).
Use the identity \(1 - \sin(2x) = (\cos x - \sin x)^2\) because \((\cos x - \sin x)^2 = \cos^2 x - 2\sin x \cos x + \sin^2 x = 1 - 2\sin x \cos x = 1 - \sin(2x)\).
Rewrite the integral as \(\int_{\pi/2}^{3\pi/4} \sqrt{(\cos x - \sin x)^2} \, dx = \int_{\pi/2}^{3\pi/4} |\cos x - \sin x| \, dx\). Consider the absolute value carefully over the interval to determine the sign of \(\cos x - \sin x\).
Split the integral if necessary based on where \(\cos x - \sin x\) changes sign, then integrate \(\cos x - \sin x\) or its negative accordingly. Use the antiderivatives \(\int \cos x \, dx = \sin x\) and \(\int \sin x \, dx = -\cos x\) to evaluate the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals
A definite integral calculates the net area under a curve between two specific limits. It involves evaluating the integral function at the upper and lower bounds and subtracting these values. Understanding the properties of definite integrals is essential for solving integrals with given limits.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Identities like sin(2x) = 2sin(x)cos(x) help simplify complex expressions inside integrals, making them easier to evaluate.
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Integration Techniques for Trigonometric Functions
Integrating functions involving trigonometric expressions often requires substitution, simplification using identities, or recognizing standard integral forms. Mastery of these techniques allows one to transform complicated integrals into solvable forms.
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Related Practice
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001
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Textbook Question
Evaluate the integrals in Exercises 33–52.
∫ cot³(t) csc⁴(t) dt
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 / ((2x + 1)√(4x + 4x²)))
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.
∫ (2^(√y) dy) / 2√y
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)
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