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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 5.5.100a
Chapter 5, Problem 5.5.100a

Substitutions Suppose ƒ is an even function with ∫₀⁸ ƒ(𝓍) d𝓍 = 9 . Evaluate each integral.                                                                                                       
(a) ∫¹₋₁ 𝓍ƒ(𝓍²) d𝓍

Verified step by step guidance
1
Step 1: Recognize that the given function ƒ is even, meaning ƒ(𝓍) = ƒ(-𝓍). This property will be useful in simplifying the integral.
Step 2: Analyze the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍. Notice that the integrand contains the term 𝓍, which is an odd function (𝓍 = -𝓍 when reflected about the origin).
Step 3: Recall the property of definite integrals: If the integrand is an odd function and the limits of integration are symmetric about zero (e.g., from -a to a), then the integral evaluates to 0. In this case, 𝓍ƒ(𝓍²) is an odd function because 𝓍 is odd and ƒ(𝓍²) is even.
Step 4: Conclude that the integral ∫₋₁¹ 𝓍ƒ(𝓍²) d𝓍 = 0 due to the symmetry of the integrand and the limits of integration.
Step 5: No further computation is needed because the integral evaluates to 0 based on the properties of odd and even functions.

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Key Concepts

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

Even Functions

An even function is defined as a function f(x) that satisfies the condition f(-x) = f(x) for all x in its domain. This symmetry about the y-axis means that the integral of an even function over a symmetric interval, such as [-a, a], can be simplified. For example, if f is even, then ∫₋ₐ⁺ₐ f(x) dx = 2∫₀⁺ₐ f(x) dx.
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Substitution in Integrals

Substitution is a technique used in integration to simplify the process of finding an integral. It involves changing the variable of integration to make the integral easier to evaluate. For instance, if we let u = g(x), then the integral ∫ f(g(x)) g'(x) dx can be transformed into ∫ f(u) du, which may be simpler to solve.
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Definite Integrals

A definite integral represents the signed area under a curve defined by a function f(x) between two limits, a and b. It is denoted as ∫ₐᵇ f(x) dx and can be computed using the Fundamental Theorem of Calculus, which states that if F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) - F(a). This concept is crucial for evaluating integrals over specific intervals.
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Related Practice
Textbook Question

Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .


(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

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Textbook Question

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (a) ∫ e¹⁰ˣ d𝓍

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Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(a) A(2)

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Textbook Question

Area functions for the same linear function Let ƒ(t) = 2t ― 2 and consider the two area functions A (𝓍) = ∫₁ˣ ƒ(t) dt and F(𝓍) = ∫₄ˣ ƒ(t) dt .

(a) Evaluate A (2) and A (3). Then use geometry to find an expression for A (𝓍) , for 𝓍 ≥ 1 .

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Textbook Question

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(a) ∫₄⁰ 3𝓍(4 ― 𝓍) d(𝓍)

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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