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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 5.4.49
Chapter 5, Problem 5.4.49

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.


∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍

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Step 1: Recall the definitions of even and odd functions. An even function satisfies f(x) = f(-x), while an odd function satisfies p(x) = -p(-x). These properties will be crucial in analyzing the symmetry of the composite function f(g(x)).
Step 2: Analyze the composition f(g(x)). Since f(x) is even, f(g(x)) will inherit the evenness of g(x) if g(x) is also even. Specifically, g(x) = g(-x) implies f(g(x)) = f(g(-x)), making f(g(x)) an even function.
Step 3: Consider the integral ∫ᵃ₋ₐ f(g(x)) dx. If f(g(x)) is even, the integral over the symmetric interval [-a, a] simplifies to 2 * ∫₀ᵃ f(g(x)) dx because the contributions from [-a, 0] and [0, a] are identical.
Step 4: If g(x) were odd, g(-x) = -g(x). Substituting this into f(g(x)), we would need to check whether f(-g(x)) = -f(g(x)) (odd) or f(-g(x)) = f(g(x)) (even). This determines whether f(g(x)) is odd or even.
Step 5: Conclude the symmetry analysis. If f(g(x)) is odd, the integral ∫ᵃ₋ₐ f(g(x)) dx evaluates to 0 because the contributions from [-a, 0] and [0, a] cancel each other out. If f(g(x)) is even, simplify the integral as described in Step 3.

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Key Concepts

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

Even and Odd Functions

Even functions are symmetric about the y-axis, meaning f(-x) = f(x) for all x. Odd functions have rotational symmetry about the origin, satisfying f(-x) = -f(x). Understanding these properties is crucial for analyzing the symmetry of composite functions and determining the nature of the integrand.
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Properties of Functions

Composite Functions

A composite function is formed when one function is applied to the result of another function, denoted as (f ∘ g)(x) = f(g(x)). In the context of the integral, knowing how to evaluate the symmetry of composite functions helps in determining whether the integrand is even or odd, which influences the integral's value.
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Evaluate Composite Functions - Special Cases

Properties of Integrals

The integral of an even function over a symmetric interval [-a, a] is twice the integral from 0 to a, while the integral of an odd function over the same interval is zero. These properties simplify the evaluation of integrals and are essential for proving the nature of the integrand in the given problem.
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Related Practice
Textbook Question

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.



∫₀ᵃ ƒ(𝓍) d𝓍

Textbook Question

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

                                                                                                                                                                              

 ∫₋₁¹ (𝓍―1) (𝓍²―2𝓍)⁷ d𝓍

Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.                                                                                  

                                                                                                                                                                    

 ∫ (sin⁵ 𝓍 + 3 sin³ 𝓍― sin 𝓍) cos 𝓍 d𝓍

Textbook Question

The linear function ƒ(𝓍) = 3 ― 𝓍 is decreasing on the interval [0, 3]. Is its area function for ƒ (with left endpoint 0) increasing or decreasing on the interval [0, 3]? Draw a picture and explain. 

Textbook Question

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.



∫₀ᶜ |ƒ(𝓍)| d𝓍

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Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(p(𝓍)) d𝓍