Textbook Question
Suppose F is an antiderivative of Ζ and A is an area function of Ζ. What is the relationship between F and A?
Ch. 5 - Integration
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 5, Problem 5.5.87
Integrals with sinΒ² π and cosΒ² π Evaluate the following integrals.
β«βΟ^Ο cosΒ² π dπ
Verified step by step guidance1
Step 1: Recognize that the integral involves cosΒ²(π). To simplify this, use the trigonometric identity cosΒ²(π) = (1 + cos(2π)) / 2.
Step 2: Rewrite the integral using the identity: β«βΟ^Ο cosΒ²(π) dπ = β«βΟ^Ο (1 + cos(2π)) / 2 dπ.
Step 3: Split the integral into two separate integrals: β«βΟ^Ο (1/2) dπ + β«βΟ^Ο (cos(2π)/2) dπ.
Step 4: Evaluate the first integral β«βΟ^Ο (1/2) dπ. This is a constant term, so it simplifies to (1/2) * β«βΟ^Ο dπ, which is the length of the interval multiplied by 1/2.
Step 5: Evaluate the second integral β«βΟ^Ο (cos(2π)/2) dπ. Since cos(2π) is an even function and the interval is symmetric about zero, the integral of cos(2π) over [-Ο, Ο] is zero. Combine the results from both integrals 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.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. Key identities include the Pythagorean identities, such as sinΒ²(x) + cosΒ²(x) = 1, and double angle formulas. These identities are essential for simplifying integrals involving sinΒ²(x) and cosΒ²(x), allowing for easier evaluation.
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Integration Techniques
Integration techniques are methods used to find the integral of a function. Common techniques include substitution, integration by parts, and using trigonometric identities to simplify the integrand. For integrals involving sinΒ²(x) and cosΒ²(x), applying the half-angle identities can transform the integrals into more manageable forms.
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06:18Integration by Parts for Definite Integrals
Definite Integrals
Definite integrals represent the signed area under a curve between two specified limits. The notation β«_a^b f(x) dx indicates the integral of f(x) from a to b. Evaluating definite integrals often involves finding the antiderivative of the function and applying the Fundamental Theorem of Calculus, which connects differentiation and integration.
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Related Practice
Textbook Question
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
β«Ο/β^Ο/Β² (cos π) / (sinΒ² π) dπ
Textbook Question
Evaluate β«βΒ² 3πΒ² dπ and β«ββΒ² 3πΒ² dπ.
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
β« 2 / (πβ4πΒ² β1) dπ , π > Β½
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Textbook Question
Use symmetry to explain why.
β«β΄ββ (5πβ΄ + 3πΒ³ + 2πΒ² + π + 1) dπ = 2 β«ββ΄ (5πβ΄ + 2πΒ² + π + 1) dπ .
Textbook Question
Area functions from graphs The graph of Ζ is given in the figure. A(π) = β«βΛ£ Ζ(t) dt and evaluate A(2), A(5), A(8), and A(12).
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