Textbook Question
7–58. Improper integrals Evaluate the following integrals or state that they diverge.
22. ∫ (from -∞ to -2) (1/x²) sin(π/2) dx
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Ch. 8 - Integration Techniques
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 8, Problem 8.3.68
67-70. Integrals of the form ∫ sin(mx)cos(nx) dx Use the following product-to-sum identities to evaluate the given integrals:
sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)]
sin(mx)cos(nx) = ½[sin((m-n)x) + sin((m+n)x)]
cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)]
68. ∫ sin(5x)sin(7x) dx
Verified step by step guidance1
Step 1: Recognize that the integral involves the product of two sine functions, sin(5x) and sin(7x). To simplify this, use the product-to-sum identity for sin(mx)sin(nx): sin(mx)sin(nx) = ½[cos((m-n)x) - cos((m+n)x)].
Step 2: Substitute m = 5 and n = 7 into the identity. This gives sin(5x)sin(7x) = ½[cos((5-7)x) - cos((5+7)x)]. Simplify the expressions inside the cosine functions: (5-7)x = -2x and (5+7)x = 12x.
Step 3: Rewrite the integral using the product-to-sum identity: ∫ sin(5x)sin(7x) dx = ½ ∫ [cos(-2x) - cos(12x)] dx. Note that cos(-2x) = cos(2x) because cosine is an even function.
Step 4: Break the integral into two separate integrals: ∫ sin(5x)sin(7x) dx = ½ [∫ cos(2x) dx - ∫ cos(12x) dx].
Step 5: Integrate each term separately. Recall that the integral of cos(kx) is (1/k)sin(kx) + C, where k is a constant. Apply this formula to both ∫ cos(2x) dx and ∫ cos(12x) dx, and combine the results to complete the solution.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Product-to-Sum Identities
Product-to-sum identities are trigonometric formulas that express products of sine and cosine functions as sums or differences of sine and cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form, making it easier to evaluate integrals.
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Verifying Trig Equations as Identities
Integration Techniques
Integration techniques refer to various methods used to compute integrals, including substitution, integration by parts, and the use of trigonometric identities. In the context of the given question, applying product-to-sum identities is a specific technique that allows for the simplification of integrals involving products of sine and cosine functions.
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Definite vs. Indefinite Integrals
Definite integrals calculate the area under a curve between two specified limits, while indefinite integrals represent a family of functions and include a constant of integration. Understanding the difference is crucial when evaluating integrals, as it affects the final result and the interpretation of the integral's meaning in a given context.
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Related Practice
Textbook Question
60–69. Completing the square Evaluate the following integrals.
65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)
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Textbook Question
9–61. Trigonometric integrals Evaluate the following integrals.
11. ∫ sin²(3x) dx
Textbook Question
90. Work Let R be the region in the first quadrant bounded by the curve y = √(x⁴ - 4)
and the lines y = 0 and y = 2. Suppose a tank that is full of water has the shape of a solid of revolution obtained by revolving region R about the y-axis. How much work is required to pump all the water to the top of the tank? Assume x and y are in meters.
Textbook Question
76. Apparent discrepancy
Three different computer algebra systems give the following results:
∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).
Explain how all three can be correct.
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Textbook Question
9–40. Integration by parts Evaluate the following integrals using integration by parts.
29. ∫ e⁻ˣ sin(4x) dx