Textbook Question
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) 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 8.3.54
Evaluate the integrals in Exercises 53–58.
∫ sin(2x) cos(3x) dx
Verified step by step guidance1
Recognize that the integral involves the product of sine and cosine functions with different arguments: \(\int \sin(2x) \cos(3x) \, dx\).
Use the product-to-sum identity for sine and cosine: \(\sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
Apply the identity with \(A = 2x\) and \(B = 3x\) to rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \int \frac{1}{2} [\sin(2x + 3x) + \sin(2x - 3x)] \, dx\).
Simplify the arguments inside the sine functions: \(\sin(5x)\) and \(\sin(-x)\), so the integral becomes \(\frac{1}{2} \int [\sin(5x) + \sin(-x)] \, dx\).
Recall that \(\sin(-x) = -\sin x\), then split the integral into two simpler integrals: \(\frac{1}{2} \left( \int \sin(5x) \, dx - \int \sin x \, dx \right)\), and proceed to integrate each term separately.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product-to-Sum Trigonometric Identities
These identities transform products of sine and cosine functions into sums or differences of trigonometric functions, simplifying integration. For example, sin(A)cos(B) = 1/2 [sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a more manageable form.
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Verifying Trig Equations as Identities
Basic Integration of Sine and Cosine Functions
Integrating sine and cosine functions involves reversing differentiation: ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these formulas allows direct integration once the integrand is simplified.
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5:53Graph of Sine and Cosine Function
Constant Multiple Rule in Integration
This rule states that constants can be factored out of integrals, simplifying calculations. For example, ∫a·f(x) dx = a ∫f(x) dx. Applying this rule is essential when coefficients appear after using trigonometric identities.
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Related Practice
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 0 to π/2 of (cot θ dθ)
Textbook Question
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀¹ (4r dr) / √(1 − r⁴)
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Textbook Question
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.
Textbook Question
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
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Textbook Question
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)
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