Skip to main content
Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.5.100a
Capitolo 5, Problema 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

Guida verificata passo dopo passo
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
1m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
6:13
Exponential Functions

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.
Video consigliato:
04:27
Substitution With an Extra Variable

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.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

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


(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

73
views
Domanda del libro di testo

Use Table 5.6 to evaluate the following indefinite integrals.                                                                                                               

                                                                                                                                                                  

 (a) ∫ e¹⁰ˣ d𝓍

78
views
Domanda del libro di testo

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)

66
views
Domanda del libro di testo

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 .

70
views
Domanda del libro di testo

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)

61
views
Domanda del libro di testo

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.

66
views