Textbook Question
41–48. Geometry problems Use a table of integrals to solve the following problems.
46. Find the area of the region bounded by the graph of y = 1/√(x² - 2x + 2) and the x-axis from x = 0 to x = 3.
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 8.3.70
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)]
70. ∫ cos(x)cos(2x) dx
Verified step by step guidance1
Step 1: Recognize that the integral ∫ cos(x)cos(2x) dx involves the product of two cosine functions. To simplify this, use the product-to-sum identity for cos(mx)cos(nx): cos(mx)cos(nx) = ½[cos((m-n)x) + cos((m+n)x)].
Step 2: Substitute m = 1 and n = 2 into the product-to-sum identity. This gives cos(x)cos(2x) = ½[cos((1-2)x) + cos((1+2)x)] = ½[cos(-x) + cos(3x)].
Step 3: Simplify further using the fact that cos(-x) = cos(x) (since cosine is an even function). This reduces the expression to cos(x)cos(2x) = ½[cos(x) + cos(3x)].
Step 4: Rewrite the integral using the simplified expression: ∫ cos(x)cos(2x) dx = ∫ ½[cos(x) + cos(3x)] dx. Factor out the constant ½: ∫ cos(x)cos(2x) dx = ½ ∫ [cos(x) + cos(3x)] dx.
Step 5: Split the integral into two separate integrals: ∫ cos(x)cos(2x) dx = ½ [∫ cos(x) dx + ∫ cos(3x) dx]. Now, integrate each term separately. For ∫ cos(x) dx, the result is sin(x). For ∫ cos(3x) dx, use the substitution u = 3x, du = 3 dx, which leads to (1/3)sin(3x). Combine these 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 the product of sine and cosine functions as a sum of sine or cosine functions. These identities simplify the integration of products of trigonometric functions by transforming them into a more manageable form. For example, the identity for sin(mx)cos(nx) allows us to rewrite the integral in a way that is easier to evaluate.
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Integration of Trigonometric Functions
Integrating trigonometric functions involves finding the antiderivative of functions like sine and cosine. The basic integrals, such as ∫sin(x)dx = -cos(x) + C and ∫cos(x)dx = sin(x) + C, are foundational. Understanding these integrals is crucial when applying product-to-sum identities, as the transformed functions will often be simpler trigonometric functions that can be integrated directly.
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Definite and Indefinite Integrals
Definite integrals calculate the area under a curve between two limits, while indefinite integrals represent a family of functions without specific bounds. In the context of the given question, recognizing whether the integral is definite or indefinite is important for applying the correct evaluation techniques and understanding the final result. The integral ∫cos(x)cos(2x)dx, for instance, is an indefinite integral that will yield a general antiderivative.
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Related Practice
Textbook Question
72. Between the sine and inverse sine Find the area of the region bound by the curves y = sin x and y = sin⁻¹x on the interval [0, 1/2].
Textbook Question
4. Is a reduction formula an analytical method or a numerical method? Explain.
Textbook Question
7–40. Table look-up integrals Use a table of integrals to evaluate the following indefinite integrals. Some of the integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table.
8. ∫ sin 3x cos 2x dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
39. ∫ x²/(100 - x²)^(3/2) dx
Textbook Question
7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
26. ∫[√2 to √2] √(x² - 1)/x dx