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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.R.21
Chapter 5, Problem 5.R.21

Properties of integrals Suppose βˆ«β‚β΄ Ζ’(𝓍) d𝓍 = 6 , βˆ«β‚β΄ g(𝓍) d𝓍 = 4 and βˆ«β‚ƒβ΄ Ζ’(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.


βˆ«β‚Β³ Ζ’(𝓍)/g(𝓍) d𝓍

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Step 1: Begin by analyzing the given information. You are provided with the values of three definite integrals: βˆ«β‚β΄ Ζ’(𝓍) d𝓍 = 6, βˆ«β‚β΄ g(𝓍) d𝓍 = 4, and βˆ«β‚ƒβ΄ Ζ’(𝓍) d𝓍 = 2. These represent the areas under the curves of Ζ’(𝓍) and g(𝓍) over specific intervals.
Step 2: Use the property of definite integrals that allows splitting the integral over an interval into subintervals. Specifically, βˆ«β‚β΄ Ζ’(𝓍) d𝓍 = βˆ«β‚Β³ Ζ’(𝓍) d𝓍 + βˆ«β‚ƒβ΄ Ζ’(𝓍) d𝓍. Substitute the known values: 6 = βˆ«β‚Β³ Ζ’(𝓍) d𝓍 + 2. Solve for βˆ«β‚Β³ Ζ’(𝓍) d𝓍, which gives βˆ«β‚Β³ Ζ’(𝓍) d𝓍 = 4.
Step 3: Recognize that the integral βˆ«β‚Β³ Ζ’(𝓍)/g(𝓍) d𝓍 involves the division of two functions Ζ’(𝓍) and g(𝓍). However, the given information only provides the integrals of Ζ’(𝓍) and g(𝓍) separately, not their quotient. This means you cannot directly compute the integral of their division using the provided data.
Step 4: State that there is not enough information to evaluate βˆ«β‚Β³ Ζ’(𝓍)/g(𝓍) d𝓍. To compute this integral, you would need either the explicit forms of Ζ’(𝓍) and g(𝓍) or additional information about their behavior over the interval [1, 3].
Step 5: Conclude that while the properties of integrals allow manipulation of sums and differences, they do not extend to the division of functions without further details. Therefore, the integral βˆ«β‚Β³ Ζ’(𝓍)/g(𝓍) d𝓍 cannot be evaluated with the given data.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Definite Integrals

Definite integrals have several key properties, including linearity, which states that the integral of a sum is the sum of the integrals, and the ability to split integrals over adjacent intervals. For example, βˆ«β‚α΅‡ f(x) dx can be expressed as βˆ«β‚α΅— f(x) dx + βˆ«β‚œα΅‡ f(x) dx for any t in [a, b]. Understanding these properties is crucial for evaluating integrals and manipulating them effectively.
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Integration of Functions

Integration is the process of finding the area under a curve represented by a function over a specified interval. The integral ∫ f(x) dx gives the accumulated value of f(x) from a to b. In this context, knowing how to evaluate integrals of specific functions and their relationships is essential for solving the given problem involving Ζ’(x) and g(x).
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Ratio of Functions in Integrals

When dealing with the integral of a ratio of functions, such as ∫ f(x)/g(x) dx, it is important to consider the behavior of both functions over the interval of integration. If g(x) is non-zero and continuous, the integral can often be evaluated using techniques like substitution or partial fractions. However, if g(x) approaches zero, the integral may be undefined or require special consideration.
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Related Practice
Textbook Question

Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer. 


βˆ«β‚€β΄ (𝓍³―𝓍) d𝓍

Textbook Question

Function defined by an integral Let H (𝓍) = βˆ«β‚€Λ£ √(4 ― tΒ²) dt, for ― 2 ≀ 𝓍 ≀ 2.

(a) Evaluate H (0) .

Textbook Question

Find the average value of Ζ’(𝓍) = eΒ²Λ£ on [0, ln 2] .

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’ and Ζ’' are continuous functions for all real numbers.

(a) A(𝓍) = βˆ«β‚Λ£ Ζ’(t) dt and Ζ’(t) = 2t―3 , then A is a quadratic function.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ’ and Ζ’' are continuous functions for all real numbers.

(c) βˆ«β‚α΅‡ Ζ’'(𝓍) d𝓍 = Ζ’(b) ―ƒ(a) .

Textbook Question

Area of regions Compute the area of the region bounded by the graph of Ζ’ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.                                              

                                                                                                                                                                                    

 Ζ’(𝓍) = 2 sin 𝓍/4 on [0, 2Ο€]

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