Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ sin(2x) cos(4x) dx
Ch. 7 - Transcendental Functions
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 7, Problem 8.6.22
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ 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 trigonometric identity to rewrite the product of sine and cosine as a sum of sines: \(\sin(A) \cos(B) = \frac{1}{2} [\sin(A+B) + \sin(A-B)]\).
Apply the identity with \(A = 2x\) and \(B = 3x\) to get: \(\sin(2x) \cos(3x) = \frac{1}{2} [\sin(5x) + \sin(-x)]\).
Simplify \(\sin(-x)\) using the odd property of sine: \(\sin(-x) = -\sin(x)\), so the expression becomes \(\frac{1}{2} [\sin(5x) - \sin(x)]\).
Rewrite the integral as \(\int \sin(2x) \cos(3x) \, dx = \frac{1}{2} \int [\sin(5x) - \sin(x)] \, dx\), then integrate each sine term separately using the integral formula \(\int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C\).
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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) = ½[sin(A+B) + sin(A−B)]. Using these identities helps convert the integral into a form easier to integrate.
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Basic Integration of Trigonometric Functions
Integrating sine and cosine functions involves applying standard integral formulas, such as ∫sin(kx) dx = −(1/k)cos(kx) + C and ∫cos(kx) dx = (1/k)sin(kx) + C. Recognizing these forms allows direct evaluation after simplification.
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Use of Integral Tables
Integral tables provide formulas for common integrals, including trigonometric integrals, which can save time and reduce errors. Referring to these tables helps identify the correct integral form and constants, especially for more complex expressions.
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Related Practice
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.
∫ (tan²x + sec²x) dx
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
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Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. (1.1)^x